Transport, Criticality, and Chaos in Fermionic Quantum Matter at Nonzero Density

Transport, criticality, and chaos in fermionic quantum matter at nonzero density

a dissertation presented by Aavishkar Apoorva Patel to The Department of Physics

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics

Transport, criticality, and chaos in fermionic quantum matter at nonzero density

Abstract

This dissertation is a study of various aspects of metals with strong interactions between electrons, with a particular emphasis on the problem of charge transport through them. We consider the physics of clean or weakly-disordered metals near some quantum critical points, and highlight novel transport regimes that could be relevant to experiments. We then develop a variety of exactly-solvable lattice models of strongly interacting non-Fermi liquid metals using novel non-perturbative techniques based on the Sachdev-Ye-Kitaev models, and relate their physics to that of the ubiquitous “strange metal” normal state of most correlated- electron superconductors, providing controlled theoretical insight into the possible mechanisms behind it.

Finally, we use ideas from the field of quantum chaos to study mathematical quantities that can provide evidence for the existence of quasiparticles (or the lack thereof) in quantum many-body systems, in the context of metals with correlated electrons.

iii Contents

0 Introduction 1 0.1 The quantum mechanics of metals ...... 1 0.2 “Strange” metals ...... 4 0.3 Metals beyond Fermi liquids ...... 6 0.4 Many-body quantum chaos ...... 14

1 Hyperscaling at the spin density wave quantum critical point in two dimensional metals 18 1.1 Introduction ...... 18 1.2 Model ...... 24 1.3 Optical conductivity σ(ω) ...... 28 1.4 T > 0 free energy ...... 37 1.5 Quantum Boltzmann Equation ...... 40 1.6 Discussion ...... 51

2 DC resistivity at the onset of spin density wave order in two-dimensional metals 53 2.1 Introduction ...... 53 2.2 Symmetries and Noether currents ...... 60 2.3 Memory Matrix Approach ...... 61 2.4 Contributions to the DC Resistivity ...... 63 2.5 Discussion ...... 68

3 Hydrodynamic flows of non-Fermi liquids: magnetotransport and bilayer drag 70 3.1 Introduction ...... 70 3.2 Hydrodynamic formalism ...... 72 3.3 Magnetotransport in a single layer ...... 73 3.4 Drag transport in bilayers ...... 76 3.5 Discussion ...... 80

4 Shear viscosity at the Ising-nematic quantum critical point in two dimensional metals 82 4.1 Introduction ...... 82 4.2 Scaling arguments ...... 85 4.3 Field theory ...... 87 4.4 Optical shear viscosity ...... 90 4.5 Boltzmann equation and DC viscosity ...... 94 4.6 Discussion ...... 102

5 Magnetotransport in a model of a disordered strange metal 103 5.1 Introduction ...... 103 5.2 Microscopic model ...... 108 5.3 Fate of the conduction electrons ...... 110 5.4 Transport in a single domain ...... 118 5.5 Macroscopic transport via Effective-medium/Random-resistor theory ...... 129

iv 5.6 Discussion ...... 136

6 Coherent superconductivity with a large gap ratio from incoherent metals 140 6.1 Introduction ...... 140 6.2 Model 1 ...... 142 6.3 Model 2 ...... 145 6.4 Discussion ...... 149

7 A critical strange metal from fluctuating gauge fields in a solvable random model 151 7.1 Introduction ...... 151 7.2 Model and large-N limit ...... 154 7.3 Single-particle properties ...... 160 7.4 Thermodynamics ...... 169 7.5 Transport ...... 174 7.6 Discussion ...... 177

8 Quantum chaos on a critical Fermi surface 179 8.1 Introduction ...... 179 8.2 Model ...... 182 8.3 Scrambling and the Lyapunov exponent ...... 185 8.4 The butterfly effect and energy diffusion ...... 191 8.5 Discussion ...... 195

9 Quantum butterfly effect in weakly interacting diffusive metals 197 9.1 Introduction ...... 197 9.2 Preliminaries ...... 202 9.3 Many-body quantum chaos ...... 205 9.4 Additional considerations ...... 214 9.5 Two spatial dimensions ...... 219 9.6 Discussion ...... 221

Appendix A Appendices to Chapter 1 223 A.1 Computation of ⟨JyJy⟩ ...... 223 A.2 Free Energy Computations ...... 233 A.3 Finite v and c ...... 238 A.4 Boltzmann Equation Computations ...... 240

Appendix B Appendices to Chapter 2 251 B.1 Susceptibilities ...... 251 B.2 Computation of R(T ) ...... 253 B.3 Random Mass Computations ...... 257 B.4 Vertex Correction for Inter Hot-Spot Scattering ...... 259

Appendix C Appendices to Chapter 3 263 C.1 Solution to linearized disordered magneto-hydrodynamic equations ...... 263 C.2 Magneto-thermal transport in the clean system ...... 267 C.3 Solution to hydrodynamic equations of the bilayer system ...... 268 C.4 Remarks on Coulomb drag in the ν = 1/2 quantum Hall state ...... 274

v Appendix D Appendices to Chapter 4 278 D.1 Optical viscosity: two-loop computations ...... 278 D.2 Relating conductivities and viscosities using Ward identities ...... 280 D.3 Contributions from the full Fermi surface ...... 285

Appendix E Appendices to Chapter 5 290 E.1 Effects of ‘Pair-hopping’ and bilinear terms on the marginal-Fermi liquid ...... 290 E.2 Boltzmann equation for the marginal-Fermi liquid ...... 292

Appendix F Appendices to Chapter 6 297 F.1 Derivation of gap equations ...... 297 F.2 Superconducting transition energetics ...... 299 F.3 Real-time Dyson equations ...... 300 F.4 Gap equations for Model 2 ...... 303

Appendix G Appendices to Chapter 7 304 G.1 Higgs transition from the U(1) ACL to a Z2 ACL ...... 304

Appendix H Appendices to Chapter 8 309 H.1 Self energies ...... 309 H.2 Wightman functions ...... 311 H.3 Higher order corrections ...... 312 H.4 Numerical methods ...... 313 H.5 Specific heat and thermal conductivity ...... 315

Appendix I Appendices to Chapter 9 318 I.1 Outline of Feynman rules for the complex-time contour ...... 318 I.2 Absence of chaos in the non-interacting disordered metal ...... 320

References 345

vi Citations to previously published work

Most of this dissertation has appeared in print elsewhere. Details are given below:

• Chapter 1 has appeared as A. A. Patel, P. Strack and S. Sachdev, Physical Review B 92 (16), 165105

(2015).

• Chapter 2 has appeared as A. A. Patel and S. Sachdev, Physical Review B 90 (16), 165146 (2014).

• Chapter 3 has appeared as A. A. Patel, R. A. Davison and A. Levchenko, Physical Review B 96 (20),

205417 (2017).

• Chapter 4 has appeared as A. Eberlein*, A. A. Patel* and S. Sachdev, Physical Review B 95 (7),

075127 (2017).

• Chapter 5 has appeared as A. A. Patel, J. McGreevy, D.P. Arovas and S. Sachdev, Physical Review

X 8 (2), 021049 (2018).

• Chapter 6 has appeared as A. A. Patel and S. Sachdev, Physical Review B 98 (12), 125134 (2018).

• Chapter 7 has appeared as A. A. Patel, M.J. Lawler and E.-A. Kim, Physical Review Letters 121

(18), 187001 (2018).

• Chapter 8 has appeared as A. A. Patel and S. Sachdev, Proceedings of the National Academy of

Sciences 114 (8), 1844-1849 (2017).

• Chapter 9 has appeared as A. A Patel, D. Chowdhury, S. Sachdev and B. Swingle, Physical Review

X 7 (3), 031047 (2017).

*Equal contributions

vii Some additional work performed by the author over the course of his graduate studies, but not included in this dissertation has appeared in

• A. A. Patel, D. Chowdhury, A. Allais and S. Sachdev, Confinement transition to density wave order

in metallic doped spin liquids, Physical Review B 93 (16), 165139 (2016).

• A. A. Patel and A. Eberlein, Light induced enhancement of superconductivity via melting of competing

bond-density wave order in underdoped cuprates, Physical Review B 93 (19), 195139 (2016).

• A. A. Patel and D. Chowdhury, Two-dimensional spin liquids with Z 2 topological order in an array

of quantum wires, Physical Review B 94 (19), 195139 (2016).

viii Acknowledgments

Almost every kitten eventually learns to hunt. Almost every physicist eventually gets a Ph.D. . However, both these occurrences are facilitated by plenty of assistance from other entities. I am grateful for all the support I have received so far throughout my ongoing journey to become a full-fledged physicist.

I would like to thank my parents Rashmi and Apoorva for setting up the initial conditions, and significant portions of the Hamiltonian that have evolved me into my current state. They have always encouraged me to take up challenges, take risks, and solve hard problems, and have inculcated in me, through great effort, the spirits of creativity and scientific inquiry from an early age, sometimes at a cost to their own careers.

Furthermore, they have been (and continue to be) my most important sources of material, emotional, and financial support. This dissertation is dedicated to them.

I have been extremely fortunate to have Subir Sachdev as my advisor. I have learnt a lot from his creativity, the breadth and depth of his knowledge, and from his penchant for illustrating beautiful physics through rigorous calculations. His insistence that I develop my own research projects and directions has helped me acquire a degree of independence that will benefit me greatly going forward. I am also grateful for his generosity in supporting my professional development by sponsoring my attendance at several conferences and by nominating me for various fellowships. I would also like to thank my other dissertation committee members: Eugene Demler and Philip Kim, for their advice and support.

Over the course of graduate school, I have had (and have some ongoing) great collaborations with An- drea Allais, Daniel Arovas, Peter Cha, Debanjan Chowdhury, Richard Davison, Andreas Eberlein, Eun-Ah

Kim, Michael Lawler, Kyungmin Lee, Peter Lunts, Alex Levchenko, John McGreevy, Philipp Strack, Brian

Swingle and Nandini Trivedi. It would be a pleasure to work with them again in the future.

My mentors during my undergraduate career: Amit Dutta, Volodya Falko, Subroto Mukerjee, Rahul Pan-

ix dit and Diptiman Sen taught me a lot of things that were helpful to my future professional development. I am particularly indebted to Volodya Falko, who offered me, then an unknown undergraduate student from a third-world country, the opportunity to work in his research group at Lancaster University in response to a random email I sent him. I believe that my successes in projects there were instrumental to the subsequent course of my career, as I did not have the near-perfect GPA that typical Indian students who manage to get into the top graduate programs in the world do.

I would particularly like to thank Lisa Cacciabaudo, who, besides being an outstanding graduate program administrator at Harvard, is an extraordinarily kind person and friend whose support made many stressful and depressing moments of graduate school more bearable. The same goes for Carol Davis. I have eaten way too much candy and other goodies in graduate school thanks to them ;-) .

My interactions with numerous other physicists over the course of my graduate career: Ehud Altman,

James Analytis, Laurel Anderson, Waseem Bakr, Leon Balents, Sumilan Banerjee, Erez Berg, Subhro Bhat- tacharjee, Nicholas Bonesteel, Hitesh Changlani, Shubhayu Chatterjee, Christie Chiu, Andrey Chubukov,

Liang Fu, Wenbo Fu, Tarun Grover, Blaise Goutéraux, Yingfei Gu, Bertrand Halperin, Sean Hartnoll, Timo- thy Hsieh, Cassandra Hunt, David Huse, Chao-Ming Jian, Valentin Kasper, Vedika Khemani, Steven Kivel- son, Igor Klebanov, Sung-Sik Lee, Leonid Levitov, Andrew Lucas, Joel Moore, Juan Maldacena, Connie

Mousatov, Chaitanya Murthy, Rahul Nandkishore, Matthew Nichols, Nirav Patel, Xiao-Liang Qi, Brad

Ramshaw, Matthew Rispoli, Richard Schmidt, Vijay Shenoy, Senthil Todadri, Ashvin Vishwanath, Yux- uan Wang, Yochai Werman, Dominik Wild, William Witczak-Krempa, Cenke Xu, Norman Yao, Jan Zaanen and Michael Zaletel, have all provided me with little bits of knowledge that together add up to something significant. I look forward to collaborating with some of them in the future.

I also thank the ever-helpful Elizabeth Alcock, Jacob Barandes, Hannah Belcher, Silke Exner and Darryl

Zeigler at Harvard. Finally, I would like to thank my graduate school friends Maryrose Barrios, Olivia Miller and Michael Rowan for their companionship. Onwards to Berkeley!

x To my parents, Rashmi and Apoorva.

xi Beginnings do not divulge their ends.

Marty Rubin

0 Introduction

0.1 The quantum mechanics of metals

To a generic member of the public, metals are shiny objects that conduct electricity very well. However, to a modern theoretical physicist, a metal is a delicate quantum many-fermion system with a large number of gapless modes that support long-range entanglement in real space. As such, interactions between fermions must in general be treated with great care while attempting to come up with a theoretical description of such a system. Most of our twentieth-century understanding of common metals in spatial dimensions d greater than one was however greatly simplified by Landau’s Fermi liquid theory, which contained the notions of

“adiabatic continuity”, and the “quasiparticle” concept [1]. With adiabatic continuity, Landau posited that,

1 Figure 1: The quasiparticle concept. Figure taken from R. D. Mattuck, A Guide to Feynman Diagrams in the Many- body Problem, Courier Corporation (1992). under the influence of interactions, the labels associated with the non-interacting single-fermion eigenstates are more robust than the eigenstates themselves: while the eigenstates get continuously deformed starting from the non-interacting ones as interactions are ramped up from zero, they can continue to be labeled by the same quantum numbers. Then, the non-interacting fermions near the Fermi energy get deformed into weakly interacting “renormalized” quasiparticles. Like the original fermions, these quasiparticles are long lived additive excitations that obey Pauli’s exclusion principle, but have different (renormalized) values of mass and velocity (Fig. 1). Due to their similarity with non-interacting fermions, many of the properties of common metals, such as a specific heat that scales linearly with the temperature T , resemble those of a non-interacting Fermi gas with a nonzero density of states at the Fermi energy. In the case of a translationally invariant metal, such a picture leads to a “Fermi sea” of quasiparticles in momentum space, bounded by a

“Fermi surface” (Fig. 2).

The many-body excitation energy of a metal with quasiparticles can further be generically written as

∑ ∑ E = ϵαδnα + Fα,βδnαδnβ + ..., (1) α α,β

where α are the non-interacting single-fermion eigenstate labels, ϵα are the single-quasiparticle energies, and

δnα are the changes in quasiparticle occupation numbers with respect to the ground state. The ... denotes higher order terms in δn.

Crucial to the stability of the Fermi liquid picture is that the quasiparticle excitations are long lived. In

2 Figure 2: The ground state of a Fermi liquid in momentum space. All the states inside the Fermi surface are filled with Landau quasiparticles. An excitation is made by promoting a quasiparticle from a state below the Fermi surface to an empty one above it. Figure taken from A. J. Schofield, Non-Fermi liquids, Contemporary Physics 40, 95-115 (1999). dimensions d > 1*, this occurs primarily because the Pauli exclusion principle generally severely limits most scattering processes near the Fermi energy EF , and additionally because Thomas-Fermi screening by a nonzero density of fermions reduces long-range Coulomb interactions to short-range interactions [3].

The interactions between quasiparticles near the Fermi energy are thus irrelevant in the renormalization- group sense, and the quasiparticle decay rate Γ ∼ max(|δE|s,T s), where s > 1, is much smaller than the quasiparticle excitation energy δE ≡ E − EF , making the quasiparticle long lived and thus well-defined.

Although a few interaction-induced breakdowns of Fermi liquid, most notably ones leading to superconductivity [3, 4] caused by pairing of quasiparticles near the Fermi energy due to effective attractive interactions induced by lattice vibrations, were studied extensively, it is only towards the end of the 20th century that the concept of metallic states beyond Fermi liquid theory in d > 1 that did not have any well-defined quasiparticles began to be appreciated. These were discovered in the context of high-temperature superconductivity (Sec. 0.2), and are still very poorly understood.

In several situations, quantum many-body systems often display emergent fermionic quasiparticles [2,

5] that can form a “metal” which is a Landau Fermi liquid of these emergent quasiparticles, even though

*In d = 1, due to the phase space restrictions on scattering by interactions being less stringent, the fermionic quasiparticles are generically not long lived. The resulting state is not a Fermi liquid, but a “Luttinger liquid” [2] that can be described in terms of emergent weakly interacting bosonic quasiparticles. In this dissertation we will always have d > 1, and will not be concerned about Luttinger liquid physics.

3 Figure 3: A sketch of the phase diagram of a generic correlated-electron superconductor. The metallic state above Tc in between the insulating and Fermi liquid phases is not a Fermi liquid, and is called a strange metal. The exact nature of the insulating state is different in different materials. Some materials have additional “pseudogap” phases in between the insulator and the strange metal, details of which are not discussed in this dissertation. There might also be a quantum critical point (QCP) near the optimal doping at which Tc is maximum that drives the strange metallic behavior above Tc. they may not carry the quantum numbers of electrons. Our focus in this dissertation will mostly be on metallic states that do not have any quasiparticles whatsoever, not even emergent ones, although sometimes the degrees of freedom which we will use to describe the systems are more suited to being emergent fermions and not electrons.

0.2 “Strange” metals

Most correlated-electron high-temperature superconductors (including very recently discovered ones), which are composed of effectively two-dimensional layers, display anomalous metallic states at temperatures above the superconducting critical temperature Tc at optimal doping [6–10] (Fig. 3). While these are legitimate metals, with a finite electrical conductivity and a continuously variable density of charge carriers, most of their properties cannot be explained by Fermi liquid theory and they are hence dubbed “strange”. Their most striking anomalous properties are detailed below:

(i) Most strange metals near optimal doping have resistivities that scale linearly with temperature, over a very wide range of temperatures ranging from temperatures much less than Tc (if Tc is suppressed by applying

4 a magnetic field as in Ref. [11]) to > 10Tc in some materials (see Ref. [12] for example). The slope of the

T -linear resistivity, dρ/dT , is remarkably constant over the entire temperature range. Fermi liquid theory generically predicts a very different T 2 temperature dependence of the resistivity, caused by quasiparticle decays arising from electron-electron Umklapp scatterings in a one-band model. A simple mechanism of

T -linear resistivity arising from electrons scattering off phonons above their Debye temperature [13] is also clearly ruled out, as the slope is insensitive to the temperature crossing the phonon Debye temperature.

(ii) The magnitude of the T -linear resistivity for each two-dimensional layer often exceeds h/e2 at relatively moderate temperatures itself [14], leading to so-called “bad-metal” behavior. In terms of a quasiparticle picture, this corresponds to the quasiparticle mean-free-path becoming smaller than a lattice constant

(violating the so-called Ioffe-Regel “limit” [15]). Therefore, at least at these temperatures, a quasiparticle description of charge carriers is incorrect.

(iii) If the DC conductivity σ in the strange metal phase is described in terms of a Drude formula,

ne2τ σ = , (2) m∗ where n is the density of carriers estimated from the doping, and m∗ is their effective mass obtained from specific heat or quantum oscillation measurements in the adjoining Fermi liquid phase, the scattering rate

1/τ is always ∼ kBT/ℏ independent of the material [10, 11, 16–20].

(iv) In some cuprates and heavy-fermion materials, the specific heat in the strange metal phase follows an anomalous CV ∼ T ln(1/T ) behavior instead of CV ∼ T of Fermi liquid theory [21, 22].

(v) The optical conductivity of some strange metals follows an unconventional power law scaling σ(ω) ∼

ω−2/3 [23].

(vi) Recently, strange metal phases in at least two different materials have been found to show linear-in- field (B) magnetoresistance at high B, with a mutual scaling between field and temperature at low tempera-

5 tures, i.e. ρ(T,B) = T f(B/T ) [17, 19].

Developing a theoretical description of strange metal phases, especially one that can capture some of their seemingly material-independent universal aspects, is a longstanding and still open challenge that necessitates the consideration of metallic phases with strong interactions between quasiparticles that potentially precludes a quasiparticle description altogether. In parts of this dissertation, we will construct and study the properties of various theoretical models of “non-Fermi liquids”: metals that do not allow a quasiparticle description, with the phenomenology of strange metals in mind, as detailed below in Sec. 0.3.

0.3 Metals beyond Fermi liquids

Given its large number of gapless modes at the Fermi energy, it is remarkably easy to destabilize a Fermi liquid by inducing interactions between the gapless fermions that are not as restricted by phase space or that can avoid being screened. In fact, from a theoretical standpoint, it is rather fortuitous that Coulomb interactions in an electron gas usually lead to a Fermi liquid and not something drastically different. One way of introducing interactions capable of destroying quasiparticles is by coupling the fermions to fluctuations of a quantum critical order parameter [24, 25]. Due to the definition of the quantum critical point being the set of parameters for which the order parameter fluctuations are exactly gapless, with any screening effects canceled by appropriate order parameter mass terms, the interactions between fermions mediated by these fluctuations are singular at low energies (some exceptions exist if the fluctuations are long-wavelength Goldstone modes coming from the spontaneous breaking of continuous symmetries whose generators commute with momentum [26]). The quasiparticle decay rate due to these interactions then becomes Γ ∼ max(|δE|s,T s), where s ≤ 1, and is now much larger than δE, so quasiparticles are no longer well-defined in the original fermion basis. Emergent fermions arising from the fractionalization of physical degrees of freedom are often coupled to dynamically fluctuating gauge fields due to redundancies in the fractionalized description. These gauge

6 fields are protected gapless modes if the gauge group is continuous, and behave like quantum critical order parameters [27–29], also leading to the destruction of quasiparticles.

Fluctuations of critical order parameters can be traditionally classified into two different kinds: (i) Long wavelength (K = 0), such as ferromagnetic fluctuations, and (ii) Finite wavelength (K ≠ 0), such as antiferromagnetic fluctuations†. While K = 0 fluctuations can destroy quasiparticles all over the Fermi surface,

K ≠ 0 fluctuations destroy fluctuations only over the fractions of the Fermi surface (known as hot regions or hot spots) which can be connected to other points on the Fermi surface by translating by K, as gapless fermion modes must be available near both q and K + q for quasiparticles to decay. We will discuss various aspects of non-Fermi liquid behavior due to K ≠ 0 critical antiferromagnetic fluctuations, which might be relevant to the putative quantum critical points in various correlated electron superconductors [30], in Chap- ters 1, 2. We also consider aspects of non-Fermi liquid behavior induced by K = 0 fluctuations associated with nematic and ferromagnetic criticality, or coupling to emergent gauge fields, that are relevant to other field theoretic descriptions of hole-doped cuprate superconductors [29], and half-filled Landau levels [31], in Chapters 3, 4, 8.

A large portion of this dissertation is devoted to the study of transport properties in non-Fermi liquid metals, particularly electrical conductivities. Many people intuitively associate the decay of charge current in metals, and hence the generation of a finite electrical resistance, with the decay of quasiparticles themselves.

This is however grossly incorrect, as the excitation of a charge current generally implies the addition of momentum to the system, which is encapsulated in the nonzero value of the current (J)-momentum (P) static susceptibility ∫ 1/T χJP = dτ⟨J(τ)P(0)⟩, (3) 0 that defines an operator overlap of J and P. Except in special particle-hole symmetric cases (such as the

† The fluctuating order parameter fields live at wavevectors K + q where |q| ≪ |K|, kF and kF is the Fermi wavevector.

7 model considered in Chapter 1), where currents can be excited without adding momentum, the nonzero momentum carried by a current cannot be lost in a translationally invariant system, and the current thus cannot decay, leading to an infinite DC charge conductivity. This has nothing to do with the existence (or lack of) quasiparticles, and will be reproduced by any correct field-theoretic approach for calculating the current correlation functions responsible for transport. For the conductivity to be finite, and for momentum to relax, continuous translational invariance must be broken, either by a lattice (which leads to Umklapp processes that relax momentum but not crystal momentum), or by disorder (which breaks translational invariance outright).

Both types of breaking will be considered in the chapters of this dissertation that deal with transport.

A particularly interesting situation occurs when translational invariance is only weakly broken, i.e. when the operators inducing umklapp processes or coupling to disorder are small relative to the momentum- conserving interaction terms at the energy scale of interest in the field theory of the metal. Then the “strong” momentum-conserving interactions, which may also destroy quasiparticle behavior, erase the independent identities of degrees of freedom localized in momentum space at the long time scales associated with DC transport. The important variable that controls transport instead becomes the slowly-relaxing total momentum P, whose dynamics can be described by treating the charge carriers in a metal as a fluid governed by a set of hydrodynamic equations [32] (Fig. 4). The terms allowed in the hydrodynamic equations must be derived in a long-wavelength low-frequency gradient expansion keeping the symmetries of the field theory in mind [33], and various coefficients appearing in them are related to correlation functions computed at shorter distance and time scales in the original field theory. The electron fluid has some similarities to water or other familiar fluids, and shows behavior such as viscosity [34]. Chapter 1 also describes non-Fermi liquid transport induced by various kinds of disorder in a hydrodynamic regime of a metal with antiferromagnetic quantum criticality. In Chapter 3, we consider how the viscous flow of two-dimensional electron fluids in a weakly disordered background is influenced by transverse magnetic fields, finding an interesting dependence of the magnetoresistance on its shear viscosity. We also study Coulomb drag between two layers of

8 Figure 4: (a) Quasiparticles scattering off impurities. Here the lifetime of a quasiparticle in a momentum eigenstate sets the transport scattering rate, as the collisions change the total momentum of the quasiparticles. (b) A hydrodynamic picture: quasiparticles undergo rapid momentum-conserving collisions with each other, leading to a small mean free path, and transport is instead described by how a “fluid” of quasiparticles flows around impurities rather than the dynamics of its individual constituents. Figure credit: Subir Sachdev and Andrew Lucas. non-Fermi liquids in the hydrodynamic regime, and point out interesting dependencies of the drag resistance on temperature and the separation between the layers. In Chapter 4, we derive the temperature dependence of the shear viscosity of a non-Fermi liquid with a Fermi surface coupled to K = 0 critical fluctuations or an emergent gapless gauge field.

If we want to be braver and consider interactions beyond the paradigm of momentum-conserving interactions between modes living near a Fermi surface, we can construct several different lattice models of non-Fermi liquid metals. At the core of this new approach lies the Sachdev-Ye-Kitaev (SYK) model. The

SYK model is a remarkably simple zero-dimensional model of a finite density of fermions interacting via random all-to-all interactions [35–37] (Fig. 5):

∑N ∑N 1 † † † H = J c c c c + µ c c . (4) SYK (2N)3/2 ijkl i j k l i i i,j,k,l=1 i=1

The fermions are all at the same point in space and carry flavor indices 1, ... , N, and the model is solved in

→ ∞ ≪ ≫ ∗ the N limit. The Jijkl are independent random couplings satisfying Jijkl = 0, Jijkl = Jlkji =

2 2 −Jjikl = −Jijlk and ≪ |Jijkl| ≫= J , where ≪ ... ≫ denotes disorder-averaging, and µ is a chemical potential.

9 Figure 5: A cartoon of the SYK model. There are N flavors or “orbitals” of fermions (points in the figure), which all interact with each other through random 4-fermion interactions (lines in the figure). Figure credit: Leon Balents and Subir Sachdev.

Many aspects of the SYK model are solvable exactly in the large-N limit. Most important to us is the

Green’s function 1 ∑N G(τ − τ ) = ⟨T c(τ )c†(τ )⟩, (5) 1 2 N τ 1 2 i=1 where Tτ denotes time-ordering. In the large-N limit, G satisfies the exact Dyson equation

1 2 2 G(iωn) = , Σ(τ1 − τ2) = −J G (τ1 − τ2)G(τ2 − τ1), (6) iωn + µ − Σ(iωn)

which is given by the diagram summation shown in Fig. 6 due to the randomness of the Jijjkl. These equations have a scale-free power-law solution at long relative times τ ≫ 1/J, which is like a correlation function of fields at a quantum critical point. It is given by G(τ) ∼ −sgn(τ)/|τ|1/2 at T = 0, and leads to the T = 0 spectral function A(ω) ∼ 1/|ω|1/2 for |ω| ≪ J that lacks the δ-function peaks that one would associate with the obvious presence of quasiparticles. This T = 0 power law scaling with the exponent of

1/2 occurs for µ ≠ 0 as well [35, 37], although there is a first-order transition to a free-fermion phase if

|µ| ≳ 0.24J [38]. Rather, at T = 0, µ ≠ 0 simply introduces an inequivalence between the magnitudes of the coefficients of the power laws for ω, τ > 0 and ω, τ < 0.

† Since the structure of the spectral function is sensitive to the operators used to define it (here ci, ci ),

10 Figure 6: Diagrammatic representation of the Dyson equation for the Green’s function of the SYK model. Here G0(ωn) = 1/(iωn + µ). The dotted blue line denotes the contraction of random couplings ≪ JijklJlkji ≫ that restricts the set of diagrams to be summed in the large-N limit to those generated by the recursion relation implicit in the above figure. one has to instead look at the many-body spectrum to truly confirm that there are no hidden quasiparticles existing in a different operator basis‡. This was done in a numerical exact diagonalization study [39], which found that the excitation spectrum above the ground state was incompatible with the generic quasiparticle form (1), which cannot support the low-energy level spacings of ∼ e−N that appear. The SYK model realizes a “metallic” phase in the sense that the charge density Q is continuously tunable as a function of the chemical potential µ [37]. At T ≠ 0, the spectral function takes on a quantum critical scaling form

A(ω) = T −1/2F (ω/T, Q) → |ω|−1/2 for J ≫ |ω| ≫ T , where the scaling function F also depends universally on Q, and can be obtained exactly [37]. Away from N → ∞, the form of A(ω) described above is still valid for ω ≳ e−N [39], or, in other words, the critical behavior of the SYK model requires the large-N limit to be taken before the low-energy limit. At energy scales ≫ J, the SYK model becomes a nearly-free fermion model, with G(iωn) ≈ 1/(iωn + µ) for |ωn| ≫ J.

While the SYK model is zero-dimensional, it is a controlled field-theoretic description of a metallic phase without quasiparticles. A natural extension to attempt to obtain non-Fermi liquid metals in higher dimensions is to consider SYK models living on the sites of a lattice, with couplings between different sites [40–42].

These models are what we call “incoherent metals”: in the range of energy scales over which there are no quasiparticles, the Green’s functions take on a spatially local form, given by the Green’s function of the ∑ ∑ ‡ 1 N † N † For example, the model given by HRM = N 1/2 i,j=1 tijci cj + µ i=1 ci ci where tij are independent random variables, also has a ci spectral function without δ-function peaks, but clearly has quasiparticles in the diagonal basis of the quadratic hamiltonian, which is not given by ci.

11 zero-dimensional SYK model:

GIM(x, τ) = δx,0GSYK(τ). (7)

Charge conduction in an incoherent metal thus takes place by tunneling between metallic reservoirs containing a large number (∼ N) of non-quasiparticle fermions localized on lattice sites, and the conductivity can have different temperature dependencies depending on the nature of the tunneling between lattice sites. The quasiparticle mean free path in an incoherent metal is thus zero by definition. Notably, if the tunneling term is quadratic in the fermion operators, the incoherent metal has a T -linear resistivity over the range of temperature scales over which there are no quasiparticles, with the magnitude of the T -linear resistivity grossly exceeding the Ioffe-Regel “limit” [42].

In Chapter 5, we explore a different route to producing non-Fermi liquid behavior in higher dimensions using SYK models as building blocks. Instead of connecting a lattice of SYK models with tunneling terms, we instead consider a separate band of itinerant electrons interacting with a lattice of SYK impurities, in the spirit of a “Kondo lattice” problem. Without the coupling, the itinerant electrons realize a Fermi liquid.

However, with specific choices of the coupling they can realize a non-Fermi liquid phase with T -linear resistivity at low temperatures. This non-Fermi liquid however doesn’t have the spatial locality of an incoherent metal, and hence external magnetic fields can induce Lorentz forces as the charges are capable of undergo- ing orbital motion. We further show, that in the presence of disorder at large length scales, we can obtain a magnetoresistance that fits the description reported by the experiments mentioned in Sec. 0.2. We also show that this non-Fermi liquid can cross over to an incoherent and bad metal which also has T -linear resistivity and a local SYK Green’s function at higher temperatures.

In some high-temperature superconductors, Tc is high enough in the strange metal region of the phase diagram, that the T -linear resistivity at temperatures slightly above Tc exceeds the Ioffe-Regel “limit” [14].

We were thus inspired to study superconducting transitions out of SYK-based incoherent metals that have

12 T -linear resistivities, which we do in Chapter 6. We induce the superconducting transition by considering an appropriate local attractive interaction that is solvable in the large-N limit [43]. We find that, unlike in the celebrated Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity arising from Fermi liquids [4], the thermal superconducting transition out of such incoherent metals is first-order in the large-N limit. However, the ratio of the zero-temperature superconducting gap ∆ to the transition temperature Tsc, far exceeds BCS theory’s value of ≈ 1.76, which is consistent with trends observed in experiments [17, 44], even though the experimentally observed transitions are continuous.

The Hamiltonian of the SYK model (4) realizes a non-Fermi liquid ground state only when there are no quadratic terms other than the chemical potential term present. In Chapter 7, we consider a different zero- dimensional all-to-all model that combines the effects of hopping and interactions to realize a non-Fermi liquid ground state. It is given by

∑N ∑M [ ] 1 † † H = − tαβeiAij f f + (MN)1/2µδαβf f , (8) FG (MN)1/2 ij iα jβ ij iα iα ij=1 αβ=1

αβ where tij are independent random variables, Aij are dynamical U(1) gauge fields, and M and N are taken to be large with M/N fixed. Like the SYK model, this model also realizes power-law fermion spectral functions at T = 0, with a quantum critical ω/T scaling form at T ≠ 0. The power law exponent x is tunable within the range 0 < x < 1/2 as a function of M/N, and, unlike the SYK model, also varies universally as a function of the charge density Q. We further consider transitions from this non-Fermi liquid state to a metallic state with quasiparticles by quenching the fluctuations of the dynamical U(1) gauge field through a Higgs condensation mechanism. This transition also reduces the low-energy fermion density of states. This can be viewed as a solvable toy model illustrating the concept behind the transition via Higgs condensation from a (not exactly solvable) model of the strange metal phase of the underdoped cuprates that has fermions coupled to dynamical gauge fields with a continuous gauge group, to the Fermi liquid like

13 pseudogap metal phase that has a reduced density of states near the Fermi energy [29].

0.4 Many-body quantum chaos

How do we know for sure whether a metal (or other quantum many-body system) truly lacks quasiparticles and doesn’t have them in some hidden basis? As we saw in Sec. 0.3, looking at spectral functions of operators is not enough. On the other hand, the fool-proof method of numerically obtaining the full many-body spectrum is almost always impossible to implement, as the Hilbert spaces of finite-dimensional quantum systems are usually too large. In this section we will shed some light on quantum field-theoretic quantities that can successfully distinguish between the presence and absence of quasiparticles in several situations.

Classical many-body systems with strong interactions between particles are generally chaotic, and a small change in initial conditions causes a large change in the state of the system at a nonzero time, with

λ t δx(t)/δx(0) ∼ e L , where λL is a classical Lyapunov exponent. The exception to this rule is when the the system is integrable, i.e., it has a number of conserved quantities equal to the number of degrees of freedom in the system. Using Poisson brackets, we can write

δx(t) = {x(t), p(0)} ∼ eλLt. (9) δx(0)

The quantum mechanical analog of a poisson bracket is a commutator, so to quantify the analog of chaos in a quantum many-body system [36, 45], one can evaluate the squared (for positive-definiteness) commutator

(or anticommutator for fermionic operators) in a thermal ensemble (β = 1/T ),

2 CWV (t) = ⟨|[W (t),V (0)]| ⟩β, (10)

where W (t) is defined in the Heisenberg picture. To avoid ultraviolet divergences that occur from plac-

14 ing operators at the same spacetime points in quantum field theories, it is more convenient to work with a regularized (anti)commutator [46, 47],

[ ] [ ] 2 reg −βH/2 −βH/2 † −βH/4 −βH/4 CWV (t) = Tr e [W (t),V (0)]e [W (t),V (0)] = Tr e [W (t),V (0)]e , (11)

which is also positive-definite. An intuitive expectation is that since a quantum many-body system with quasiparticles (in any operator basis) has a thermodynamic number of conserved (or nearly-conserved) quantities given by the eigenvalues nα of the occupation number operators nˆα of the quasiparticle states which commute (or nearly commute) with each other at all times. The temporal growth of the (anti)commutator of generic operators W, V for which [W (0),V (0)] = 0, and which overlap with the nˆα will hence be restricted as W, V can be expressed as a sum of terms involving products of the nˆα, which will all commute with each other. As a corollary, if there are no quasiparticles and hence a non-thermodynamic number of conserved

reg quantities, CWV (t) is expected to grow rapidly in time for generic W, V for which [W (0),V (0)] = 0.

If the correlation functions ⟨V (0)W (t)W (t)V (0)⟩ and ⟨W (t)V (0)V (0)W (t)⟩ factorize as

⟨ ⟩⟨ ⟨ ⟩⟨ ⟩ reg V (0)W (t) W (t)V (0) and W (t)V (0) V (0)W (t) respectively for some range of times, then CWV (t) obeys a bound on its growth rate during that range of times:

reg d ln C (t) kBT WV ≤ 2π . (12) dt ℏ

Therefore we intuitively expect that the closer a system is to having no well-defined quasiparticles, the closer the growth rate is to saturating the bound. On the other hand, the unregulated object CWV (t) does not obey this bound [48], and its dynamics are less meaningful for gauging the strength of quasiparticle interactions [49]. The SYK model (which has no quasiparticles) saturates this bound with

∑N [ ] ( ) 1 † † 1 λ = 2πkB T t ℏ Tr e−βH/2{c (t), c (0)}e−βH/2{c (t), c (0)}† ∼ e L ℏ , t ≲ ln N. (13) N 2 i j i j N k T i,j=1 B

15 Note that the large value of N is important for having a clearly delineated and sustained regime of exponential growth. Other large-N theories, which are weakly coupled and have quasiparticles, also have this pattern of exponential growth (with the exponent λL called a Lyapunov exponent), but with parametrically smaller growth exponents λL that are suppressed by 1/N or scale as higher powers of temperature [50–52]. In these

reg theories, the growth of CWV (t) can further be tied to a Boltzmann-like equation that tracks quasiparticle collisions [49, 50]. Theories without quasiparticles are expected to have λL ∼ O(1) × kBT/ℏ, in line with old conjectures about their “dephasing” and “thermalization” rates [53].

reg If the operators are additionally separated spatially by a displacement x, CWV (x, t) exhibits characteris-

reg −| | ∼ 2πT (t x /vB ) tics of a traveling wavefront, with CWV (x, t) e /N in some lattices of coupled SYK mod-

§ els [40, 41], and in holographic gravitational theories [54–57], where vB is called the “butterfly velocity” .

∼ 2 Additionally, in many of these examples, the energy diffusion constant is DE vB/λL.

reg In Chapter 8 we study the dynamics of CWV (x, t) in a non-Fermi liquid metal consisting of an open Fermi surface coupled to a dynamical U(1) gauge field, which doesn’t have quasiparticles. This theory doesn’t have

reg → a controlled large-N limit, and CWV (x 0, t) does not have a regime of sustained exponential growth.

However at intermediate length and time scales, we find

− 2 1 (x vB t) reg λLt − C † (x, t) ∼ √ e e 4Dt , (14) ψψ Dt

∼ O × ℏ ∼ 2 ∼ where λL (1) kBT/ . Additionally, the relationship DE vB/λL D also holds in this regime.

In Chapter 9 we turn to chaos in interacting, disordered Fermi liquid metals. Here quasiparticles undergo two types of scatterings: (i) off the disordered potential, that just changes the basis of quasiparticles and doesn’t destroy them and (ii) off each other through interactions, that gives them a finite lifetime. We find

§See https://en.wikipedia.org/wiki/Butterfly_effect to understand the origin of the term.

16 that, at intermediate length and time scales,

1 |x|2 reg λLt − C † (x, t) ∼ e e 4Dt , (15) ψψ (Dt)d/2

in d > 1 spatial dimensions. The Lyapunov exponent λL turns out to be sensitive to only interaction-induced scatterings, correctly differentiating the processes that conserve quasiparticles and the ones that cause them √ to decay. The energy diffusion constant DE ∼ D, and vB can be defined as vB = 2 DλL. Furthermore in d = 2, λL is universally related to temperature and the measurable sheet resistivity of the metal, and the conditions for the saturation of the bound (12) predict the correct order of magnitude of parameters for which an experimentally observed metal-insulator transition occurs, at which quasiparticles are presumably destroyed.

17 Science is an organized pursuit of triviality...

Vera Nazarian

1 Hyperscaling at the spin density wave quantum

critical point in two dimensional metals

1.1 Introduction

The anomalous properties of the ‘strange metal’ phase of the cuprates, and other correlated electron compounds, have remained a long-standing challenge to quantum many-body theory. Strange metals are states of quantum matter whose density can be continuously varied by an external chemical potential at zero temperature, but unlike in a Fermi liquid, there are no long-lived quasiparticle excitations. It is generally believed that strange metals should be described by a strongly-coupled quantum-critical theory [8], but such a proposal

18 immediately faces an obstacle. Almost all strongly-coupled quantum-critical states, including all conformal field theories, obey the ‘hyperscaling’ property [58]: this implies that the specific heat, CV , and the conductivity, σ, scale as

d/z (d−2)/z CV ∼ T ; σ(ω ≫ T ) ∼ ω , (1.1) where ω is frequency, T is temperature, d is the spatial dimension, and z is the dynamic critical exponent; we will refer to the conductivity in the ω ≫ T regime above as the optical conductivity. In the important spatial dimension of d = 2, this immediately implies that the optical conductivity should be frequency independent, which contradicts the ∼ ω−0.65 behavior observed in the cuprates [23, 59].

The scaling arguments can also be naively extended to the DC conductivity, which would then imply that

σ(ω ≪ T ) ∼ T (d−2)/z. In d = 2, this contradicts the widely observed ‘linear-in-T resistivity’, σ ∼ T −1.

However, DC transport co-efficients are sensitive to constraints from momentum conservation, and so the naive application of hyperscaling to DC transport is often not valid [60–69]. But this sensitivity does not extend to the optical conductivity, and so the observations of Ref. [23, 59] are the stronger challenge to the hyperscaling property.

There is a much-studied [27, 28, 70–82] strongly-coupled quantum-critical point which violates hyperscaling: this is the critical point to the onset of Ising-nematic order in a metal in d = 2. A closely-related critical theory applies to a d = 2 metal coupled to an Abelian or non-Abelian gauge field. We write the properties of the Ising-nematic theory in a suggestive form similar to Eq. (1.1)

dt/z (dt−2)/z CV ∼ T ; σ(ω ≫ T ) ∼ ω , with hyperscaling violation, (1.2)

where z = 3/2 is the ‘fermionic’ dynamic critical exponent (in the notation of Ref. [77]). For the specific heat, the hyperscaling-violating dimensionality dt = 1 has been connected to the number of dimensions transverse to the Fermi surface [80, 83]. This value of dt also happens to yield the correct behavior of the

19 Figure 1.1: (a) Hot spot geometry, labelling conventions, and choice of x, y-coordinate system in the Brillouin zone of the two-dimensional square lattice in which the fermions move. The boundary of the blue area denotes the Fermi surface separating the filled particle-like states (blue area) from the the hole-like states (white area). QAF = (π, π) is the (commensurate) antiferromagnetic ordering vector that intersects the Fermi surface at 4 pairs of hot spots. (b) (Top) Fermi surface patches from a hot spot pair connected by QAF centered at a common origin in momentum space. The two light colored regions are the regions occupied by fermions at the two hot spots of the pair respectively, the dark colored region is occupied by fermions at both hot spots, and the white region is unoccupied. The arrows perpendicular to the Fermi surfaces denote the directions of the Fermi velocities. (Bottom) Under the RG flow, the Fermi surfaces are deformed as shown at the strange metal fixed point, and as indicated in Eq. (1.3). The Fermi velocities are exactly antiparallel only at the hot spot (k = 0).

optical conductivity in Eq. (1.2), although the existing [27, 84] physical interpretations of this result are

different. It is also notable that σ ∼ ω−2/3 is close to the experimental observations [23, 59].

The above violation of hyperscaling is in a theory with a ‘critical Fermi surface’. On the other hand,

theories with Dirac fermions, which are gapless only at points in the Brillouin zone do obey hyperscaling.

Our interest in this chapter is the onset of spin density wave order in two-dimensional metals, whose

critical theory is described by isolated points called ‘hot spots’ which are connected to a gapless Fermi

surface (see Fig. 1.1). This transition is therefore intermediate between the critical Fermi surface and critical

Fermi point cases. Its field theory [85] has a bosonic order parameter ϕ coupled to fermionic excitations at

4 pairs of hot spots around the Fermi surface.

20 In a large N analysis of such a field theory, it was found [85, 86] that at the two loop level that the

Fermi surfaces near the hot spots became asymptotically nested at low energies. In terms of momenta kx, ky measuring deviations from the hot spots, the Fermi surface is given by (see the bottom panel of Fig. 1.1b)

kx ky ∼ . (1.3) ln(1/|kx|)

The optical conductivity of the hot spots was computed by Hartnoll et al. [84] in a Eliashberg framework, and they found (at variance with an earlier treatment by Abanov et al. [87], and that in Ref. [88]) a hot

r spot contribution σ(ω) ∼ ω 0 , where the exponent r0 > 0 was determined by the angle between the Fermi surfaces at the hot spots. For the asympotically nested Fermi surface in Eq. (1.3), it was found [84] that r0 → 0, indicating that the optical conductivity is a constant (up to logarithms), and so obeys hyperscaling as in Eq. (1.1) in d = 2.

this chapter will re-examine these issues using the fixed point for the spin density wave critical found by

Sur and Lee [89] using an expansion in ϵ = 3 − d. They also also found the asymptotically nested Fermi surfaces in Eq. (1.3) under the 1-loop renormalization group flow of the ϵ expansion. We will review their

RG analysis in Section 1.2. We then proceed to a computation of the optical conductivity in Section 1.3, and find that the hot spot contribution obeys the hyperscaling of Eq. (1.1) (up to logarithmic corrections) in the

ϵ expansion, in agreement with Hartnoll et al. [84]. We turn to a computation of the non-zero temperature free energy density in the ϵ expansion in Section 1.4. We find a result for the hot spot contribution to the specific heat again in agreement with the hyperscaling of Eq. (1.1), and for reasons similar to those for the optical conductivity.

Sections 1.3 and 1.4 also examine the optical conductivity and the free energy in the limit of a vanishing bare ϕ velocity: c → 0. As the bare velocity is generically finite, such a limit can only apply to observable properties over intermediate ω or T : we find the allowed range is cΛ < ω, T < vF Λ, where vF is a

21 Fermi velocity (see Eq. (1.7)), and Λ is high momentum cutoff. Only in such a limit do we find hyperscaling violation as described by Eq. (1.2) with dt = 1. The quantum critical optical conductivity studied in Refs. [87,

88] is analogous to this intermediate regime, and we maintain that their results do not apply when the the bare velocity c is not small.

The more subtle question of the DC conductivity is examined in Section 1.5; in discussions of the DC conductivity, we implicitly assume that ω ≪ T . Here, we have to consider the interplay between the hot spots on the Fermi surface with the remainder of the ‘cold’ Fermi surface more carefully [66, 84, 90, 91]. The cold fermions can short-circuit electronic transport, and so possibly dominate the DC conductivity. More generally, this belongs to a class of effects associated with the conservation of total momentum, which can relax only via quenched disorder or umklapp scattering beyond that already continued in the continuum theory [66]. A general framework for describing such effects was presented in Refs. [60, 69], using solvable holographic models, relativistic hydrodynamics, and memory functions. In the context of strange metals, it useful to begin with a microscopic model in which total momentum is exactly conserved [62, 66]. Then the conductivity can be written as [60] Q2 1 σ = σ + , (1.4) Q M (−iω)

where σQ is a finite and T -dependent ‘quantum critical’ conductivity, and the second term can be viewed as the contribution of the cold Fermi surface. The pole at ω = 0 has a co-efficient determined by static thermodynamic susceptibilities associated with the electric current J and the momentum density P , with

Q = χJP and M = χPP . These thermodynamic susceptibilities are usually non-critical, and so can be taken to be non-universal and T -independent constants, which depend on the full short-distance structure of the theory. Now we add perturbations associated with umklapp scattering or quenched disorder which can relax the total momentum [60–69, 92–96]: this leads to a momentum relaxation rate Γ which shifts the pole

22 in Eq. (1.4) off the real axis to ω = −iΓ, and so the conductivity takes the finite value at ω = 0

Q2 1 σ = σ + . (1.5) Q M (−iω + Γ)

Note that Γ does have a singular T dependence associated with universal properties of the quantum-critical theory, and can be computed via memory functions [60–62, 66, 92, 97–100]. A notable feature [101] of

Eq. (1.5) is that the quantum-critical σQ and the momentum-mode conductivity are additive; this is in contrast to the Matthiessen’s Rule for quasiparticle theories, in which different quasiparticle scattering mechanisms are additive in the resistivity. The T dependence of the momentum-mode term in Eq. (1.5) was discussed in

Ref. [66], using the assumption that the cold regions of the Fermi surface are ‘lukewarm’ i. e. the electron- electron scattering rate on the entire Fermi surface is faster than the impurity scattering rate; the results of

Ref. [66] are not modified by the analysis of the present chapter.

Section 1.5 will present a computation of the quantum-critical conductivity σQ for the case of a spin density wave quantum critical point in a metal in d = 2. The momentum mode contribution in Eq. (1.5) was computed in a previous work [66], and will not be addressed here. The computation of σQ here is aided by the fact that the theory describing the hot spots is particle-hole symmetric. This implies that the scaling limit theory has Q = 0, and so we can cleanly separate away the momentum mode contribution; the full theory ultimately has Q ̸= 0, but this arises from portions of the Fermi surface away from the hot spots [66]. Such a separation between σQ and the momentum mode is more complicated in general [67]: in particular, for the Ising-nematic critical point there is no particle-hole symmetry to aid us, and we are not aware of any computation of σQ for this case. For the spin density wave critical point, we compute σQ in Section 1.5 using a Boltzmann equation method developed for conformal field theories [102–105]. We will carry out the Boltzmann analysis directly in d = 2, rather than the technically more cumbersome ϵ expansion. Consequently, our results for σQ will be qualitative, and not systematic. From the computations

23 in Section 1.5, we estimate that the leading T -dependence of σQ has the same form as the ω-dependence of the optical conductivity: i.e. with bare velocities finite, hyperscaling is preserved with σQ ∼ constant; and

(dt−2)/z with vanishing bare velocities, there is violation of hyperscaling with σQ ∼ T and dt = 1 and over intermediate T range cΛ < T < vF Λ.

1.2 Model

In this section, we first recapitulate the low-energy continuum quantum field theory for fermions moving in a two-dimensional square lattice close to the transition to the antiferromagnetic phase with commensurate ordering wave vector QAF = (π, π) [85, 86, 89]. We then explain the embedding by Sur and Lee [89] of the two-dimensional system into a higher-dimensional d = 3 − ϵ space, and summarize the basic features of the

ϵ expansion.

We begin by defining the action in frequency and momentum representation S[ψ,¯ ψ, ϕ] in two space dimensions x and y and one temporal (imaginary time) direction τ:

∫ ∫ ∑4 ∑ ∑ 1 [ ] S[ψ,¯ ψ, ϕ] = ψ¯(m)(k)[ik + em(k)] ψ(m)(k) + ϕ(−q) · q2 + c2q2 + r ϕ(q) ℓ,σ τ n ℓ,σ 2 τ ℓ=1 m= σ=↑,↓ k q ∑4 ∑ ∫ ∫ [ ] (+) (−) · ¯ ′ + g ϕ(q) ψℓ,σ (k + q)τσ,σ ψℓ,σ′ (k) + h.c. . (1.6) ℓ=1 σ,σ′=↑,↓ k q

Here, the functional integral for the fermions goes over fermionic Grassmann fields ψ¯, ψ which carry additional labels according to their “home” hot spot (depicted in Fig. 1.1). Via a “Yukawa” coupling g, the fermions are (strongly) coupled to a bosonic vector field with three components ϕ whose fluctuations represent spin waves. At zero temperature kτ is a continuous (imaginary) frequency variable with k = (kτ , k) =

(kτ , kx, ky) and likewise for q.

24 · According to Fig. 1.1, the dispersions of the fermions eℓ (k) = vℓ k in the hot regions are

− e1 (k) = e3 (k) = vF (vkx ky)

− ∓ e2 (k) = e4 (k) = vF ( kx + vky) , (1.7)

with v being the ratio of the velocities in x and y-direction; we will henceforth set vF = 1. In particular, the limit v → 0 corresponds to locally nested pairs of hot spots, in which the Fermi line becomes orthogonal to the antiferromagnetic ordering vector QAF and the fermion becomes one-dimensional and disperses parallel to QAF.

The physics of the action Eq. (1.6) in two space dimensions has been addressed with a variety of techniques including resummation of subclasses of Feynman diagrams [85], field-theoretic renormalization group techniques [84, 86], and Polchinski-Wetterich flow equations for the effective action [106]. The bottom line is that the fermions and spin-waves are strongly coupled, one has to account for strong renormalization of the shape of the Fermi surface [86].

Here we embed the fermionic system in two space dimensions described by Eq. (1.6) into a higher- dimensional space; the “extra dimensions” are added perpendicular to the physical Fermi surface [89] that lies in the x-y plane and has co-dimension 1. Artificially introduced Fermi surfaces with co-dimension > 1 are gapped out by assuming a p-wave charge density wave order in directions perpendicular to the physical

Fermi surface. This results in line nodes of the fermionic dispersion with co-dimension 1 as needed. The main advantage of this embedding is that the density of states at the Fermi line is suppressed to ρ(E) ∼ Ed−2, that is, it vanishes with energy for d > 2. This allows the powerful dimensional regularization techniques of relativistic systems to be adapted to the present problem.

25 The d + 1-dimensional action

N ∫ ∫ ∑4 ∑Nc ∑f [ ] 1 2 2 2 S = Ψ¯ (k)[iΓ · K + iγ − ε (k)] Ψ (k) + |Q| + c q Tr [Φ(−q)Φ(q)] n,σ,j d 1 n n,σ,j 4 n=1 σ=1 j=1 k q N ∫ ∫ (3−d)/2 ∑4 ∑Nc ∑f [ ] gµ ¯ + √ Ψn,σ,j(k + q)Φσ,σ′ (q)iγd−1Ψn,σ¯ ′,j(k) + h.c. (1.8) N f n=1 σ,σ′=1 j=1 k q

is integrated over k = (K, k), which contains the physical momentum k = (kx, ky) and a d − 1 = 2 − ϵ

¯ dimensional “generalized frequency” vector K = (kτ , K) = (kτ , k1, ..., kd−2), that includes the physical frequency kτ in its first component and the d − 2 extra dimensions in the others and likewise for q. The

∑ 2− [ ] Nc 1 a a a b ab bosons have been promoted to matrix fields ϕ(q) = a=1 ϕ (q)τ with Tr τ τ = 2δ conventions for

a the trace over SU(Nc) generators τ . The fermions are collected in a SU(Nf ) flavor group and the physical limit of Eq. (1.8) is

K → kτ , ϵ → 1 , d → 2 ,Nc = 2 ,Nf = 1. (1.9)

Computations with Eq. (1.8) involve traces over products of d − 1 dimensional gamma matrices, collected

¯ in the vector (Γ, γd−1) with Γ = (γ0, Γ) = (γ0, γ1, ..., γd−2), that satisfy {γµ, γν} = 2Iδµν and Tr I = 2.

The book-keeping indices for the hot spots are: 1¯ = 3, 2¯ = 4, 3¯ = 1, 4¯ = 2; the two-component fermion spinors appearing in Eq. (1.8) disperse according to

+ ε1(k) = e1 (k),

+ ε2(k) = e2 (k),

− ε3(k) = e1 (k),

− ε4(k) = e2 (k), (1.10)

26 with the right-hand-sides defined in Eq. (1.7). The two-component spinors of Eq. (1.8) contain two of the original fermions from opposing sides of the Fermi surface [89].

Sur and Lee [89] performed a field-theoretic one-loop renormalization group analysis of Eq. (1.8) in d = 3 − ϵ dimensions. They retained the simplest set of 5 independent running couplings. For the fermion propagator 3 wave-function renormalization factors are used, one in the direction of “time and extra dimensions” K and one each in the kx and ky directions. For the Bose propagator there are 2 wave-function renormalization factors, one in the Q direction and one for the qx,y directions (which have to be equivalent by point group symmetry).

The fixed point of the ϵ expansion is defined in terms of the ratios λ = g2/v and w = v/c:

N 2 + N N − 1 λ → λ∗ = 4πϵ c c f , 2 − Nc + NcNf 3 N N w → w∗ = c f , (1.11) 2 − Nc 1

The fixed point determines a dynamic critical exponent z via

λ∗ z = 1 + + O(ϵ2). (1.12) 8π

¯ z − Note that kτ and K scale as ky; so the extra d 3 spatial dimensions and the time dimension both scale as z with respect to the two physical dimensions, instead of just the time dimension as is usually the case with other models.

While the scaling structure described so far is conventional, there are logarithmic corrections which arise

27 from the flow of the velocities v and c flow to zero at long length scales. This flow is described by

dc 4z ≈ (z − 1)c2, d ln µ π dv dc ≈ w∗ , (1.13) d ln µ d ln µ where µ is the renormalization group momentum scale. Such a dynamic nesting with v → 0 was found also in an earlier 1/Nf expansion [86]. At the fixed point with vanishing v and c, the antiferromagnetic ordering vectors intersect the Fermi surface at a right angle. This is illustrated in Fig. 1.1(b). Note that with v → 0 at the fixed point, we must also have g2 → 0 for the coupling λ to remain finite; this is indeed found to be the case in the renormalization group flow.

1.3 Optical conductivity σ(ω)

In this section, we compute the optical conductivity σ(ω) for fermions near the hot spots at the ϵ expansion fixed point described in Section 1.2. Our computation will be to order ϵ, which requires evaluation of two- loop Feynman graphs.

Before embarking on the description of the Feynman graphs, let us review the expectations of a general scaling analysis. The spatial directions, x, y, have scaling dimension 1, the time direction has scaling dimension z, and the 1 − ϵ extra spatial directions with Dirac dispersion also have scaling dimension z. So the scaling dimension of the free energy density, F , is

[F ] = 2 + (2 − ϵ)z. (1.14)

The vector potential, A, has dimension 1, and so the electric current, J, being proportional to δF/δA has

28 dimension

[J] = 1 + (2 − ϵ)z. (1.15)

Finally, the conductivity is given by the Kubo formula in terms of a current correlator, from which we deduce

[σ] = (1 − ϵ)z. (1.16)

These are the scaling expectations for a theory that obeys hyperscaling. If we have a violation of hyperscaling, we expect the spatial direction along the Fermi surface to not contribute in the counting of scaling dimension.

So we should have

[F ] = 1 + (2 − ϵ)z , [σ] = −1 + (1 − ϵ)z, with hyperscaling violation. (1.17)

We already know that the Fermi liquid contribution of the quasiparticles far from the hot spots violates hyperscaling as in Eq. (1.17) with z = 1. The question before us is whether the hot spot contribution preserves hyperscaling as in Eqs. (1.14) and (1.16), or violates hyperscaling as in Eq. (1.17).

We first compute the one-loop (free fermion) contribution to the two-point correlator of the current density

⟨JyJy⟩. Then, we compute the two-loop (interaction) contributions to ⟨JyJy⟩, of which there are two: the

“self-energy correction” (Section 1.3.2) and the “vertex correction” (Section 1.3.3). Finally, in Section 1.3.4 we compile the results from the evaluation of the loop diagrams and, applying the Kubo formula, we derive the scaling form of the optical conductivity σ(ω) for the fermions near the hot spots.

29 Figure 1.2: Feynman graphs for the current-current correlator up to two loops. The black and grey boxes are current vertices for the n and n¯ hot spot pairs respectively. The wiggly lines are the boson propagators and the solid lines stand for fermion propagators. (a): One-loop contribution for free fermions. (b): Two-loop self-energy correction computed in Section 1.3.2. There is also a partner diagram with the boson on the lower fermion line. (c): Two-loop vertex correction computed in Section 1.3.3.

1.3.1 One-loop contribution to ⟨JyJy⟩

We have, for the current density in the y direction,

(1) (3) (2) (4) Jy = Jy + Jy + Jy + Jy =

∑Nc ∑Nf ∑Nc ∑Nf i (Ψ¯ 1,σ,jγd−1Ψ1,σ,j − Ψ¯ 3,σ,jγd−1Ψ3,σ,j) + iv (Ψ¯ 2,σ,jγd−1Ψ2,σ,j + Ψ¯ 4,σ,jγd−1Ψ4,σ,j), σ=1 j=1 σ=1 j=1

(1.18)

and likewise for Jx but with (1, 3) ↔ (4, 2).

The one-loop contribution to this correlator is simply the non-interacting “bubble” containing a convolution of two fermion propagators as shown in Fig. 1.2, both from the same hot spot (we follow the index convention of Eq. (1.10) and absorb the identical contributions from the other hot spots into a prefactor)

∫ 2 2−ϵ 2 d k d K ⟨J J ⟩ − (ω) = −2(1 + v )N N Tr [iγ − G (K, k)iγ − G (K + W, k)] , y y 1 loop c f (2π)2 (2π)2−ϵ d 1 1 d 1 1

(1.19)

where W = (ω,0) and the fermion propagator is given by

30 · − Γ K + γd−1εn(k) Gn(K, k) = ( i) 2 2 . (1.20) K + εn(k)

We evaluate Eq. (1.19) using Feynman parameters in App. A.1.1 and obtain in Eq. (A.2) to leading order in

ϵ:

√ ∫ ( ) 1−ϵ 1 ⟨J J ⟩ − (ω) = − 1 + v2 dk∥ N N ω . (1.21) y y 1 loop c f 16π

where k∥ is the component of k along the Fermi surface of ε1(k) (note k∥ = kx for v = 0). For comparison with the subsequent two-loop contribution, it is useful to write this as

∫ ( ) 2 dε3 1−ϵ 1 ⟨J J ⟩ − (ω) = −(1 + v ) N N ω , (1.22) y y 1 loop 2v c f 16π

where the variable of integration ε3 is a co-ordinate orthogonal to the equal energy lines of ε3(k).

We can evaluate the integral over ε3 to yield a factor of Λ, a large-momentum cutoff, and then we conclude

−ϵ that σ1−loop(ω) ∼ ω . We now observe that this result agrees with hyperscaling violating scaling dimension in Eq. (1.17) for z = 1. This is just the expected result, because we are dealing with the contribution of free fermions, and there is no distinction yet between the hot-spot contribution, and the Fermi liquid contribution of quasiparticles far from the hot spot.

31 Figure 1.3: Key one-loop elements appearing in the two-loop self-energy correction (a) and two-loop (current) vertex correction (b).

1.3.2 Two-loop self-energy correction ⟨JyJy⟩SE

To investigate the impact of interactions on ⟨JyJy⟩, we first compute the two-loop self-energy correction depicted in Fig. 1.2(b). There are two diagrams here with identical contributions, whose sum gives

∫ [ 2 2−ϵ 2 d k d K ⟨J J ⟩ (ω) = −4(1 + v )N Tr iγ − G (K, k)Σ (K, k)G (K, k) y y SE f (2π)2 (2π)2−ϵ d 1 1 1 1 ]

× iγd−1G1(K + W, k) (1.23)

Here we note that this expression contains three fermion propagators from the same hot spot pair 1 and one, inside the one-loop self-energy Σn(K, k), depicted in Fig. 1.3 (a) from its “partner” hot spot pair 3. We now compute Σ1(K, k) separately. After that, we substitute the result back into Eq. (1.23) and perform the remaining integrations over k and K. The effect of large momentum-transfer scattering of fermions from one hot spot pair (1) to its partner (3) via exchange of bosonic spin fluctuations is captured in the self-energy

2− ∫ 2 ϵ N∑c 1 2 2−ϵ g µ j j d q d Q Σ (K, k) = (τ τ ) iγ − G (Q + K, q + k)iγ − D(Q, q) , (1.24) 1 N (2π)2 (2π)2−ϵ d 1 3 d 1 f j=1 where the spin fluctuation propagator

1 D(Q, q) = (1.25) Q2 + c2q2

32 involves the spin-wave velocity c which vanishes at the Sur-Lee fixed point near the hot spots, as does the

Yukawa coupling g2; the ratio g2/c however attains a finite value (see Sec. 1.2).

We evaluate the expressions first in Appendix A.1.2 using a simplifying approximation valid only for small bare velocities c and v. In the limit v, c → 0, the integrand in Eq. (1.24) then depends on qx only via the spin fluctuation propagator. Thus, we can first perform the qx integration and then set c = 0, which is equivalent to replacing

∫ ∫ 2 d q → dqy (2π)2 (2π)2 π 1 D(Q, q) → . (1.26) c |Q| in Eq. (1.24). This way, Eq. (1.24) picks up the finite prefactor g2/c and the integrand becomes independent of both velocities v and c. The resulting integrals are performed in Appendix A.1.2 using Feynman parameters to obtain

2− ∫ − Nc 1 1 − ϵ 2 ϵ/2 ϵ 2 ∑ · 1 − 2 2 π Γ(ϵ/2) µ g j j Γ K (1 x) Σ1(K, k) = Σ1(K) = −i (τ τ ) dx . (1.27) (2π)4−ϵ N c [K2]ϵ/2 1 + ϵ f j=1 0 x 2 2

We observe that the above fermion self-energy depends only on frequency and not an spatial momenta k any- more; the fermions near the hot spots see an essentially “local” boson with the “1 over frequency” propagator of Eq. (1.26). Eq. (1.27) induces an anomalous scaling for the K dependent part of the fermion propagator, but the absence of anomalous dimensions for the spatial k components renormalizes the dynamical exponent z to values larger than one at the Sur-Lee fixed point.

However, for our purposes here, the approximation associated with Eq. (1.26) turns out not to be sufficient, since v and c vanish only logarithmically near the hot spot. It is thus crucial to obtain the full v and c dependence of the pole term in Eq. (1.27), and of that in Eq. (1.23). The needed integrals are computed in

Appendix A.3, and the final result for the two-loop self-energy correction is the rather complicated expression

33 in Eqs. (A.38) and (A.39). Computing its singular pole in ϵ and dropping power divergent terms, we obtain

∫ ∫ dε (N 2 − 1)g2µϵω1−ϵ 1 (1 − x)1/2(1 + v2) ⟨J J ⟩ (ω) ≈ 3 c dx y y SE 3 2 2 2 1/2 2v 64π cϵ 0 (c + x(1 + v − c )) ( )− c2ε2 ϵ/2 ω2 + 3 (1.28) c2 + x(1 + v2 − c2)

in terms of the same ε3 variable of integration used in Eq. (1.21).

1.3.3 Two-loop vertex correction to ⟨JyJy⟩vert

The vertex correction graph, Fig. 1.2(c), is considerably more involved than the self-energy correction of the preceding section; although for c → 0 it is free of 1/ϵ poles of the type Eq. (1.28). Using the abbreviation

Ξ1(K, k, W) for the one-loop current vertex correction in Fig. 1.3(b), we can write the entire graph including contributions from all hot spot pairs as

∫ ∫ [ ] 2 2−ϵ 2 d k d K ⟨J J ⟩ (ω) = −2(1 − v )iN Tr γ − G (K, k)Ξ (K, k, W)G (K + W, k) , y y vert f (2π)2 (2π)2−ϵ d 1 1 3 1

(1.29)

We observe that Eq. (1.29) contains two fermion propagators from one hot spot pair and two fermion propagators (inside the current vertex Ξ1) from the partner hot spot pair, unlike the self energy correction Eq. (1.23).

The one-loop correction to the Jy vertex,

[ 2− ∫ 2 ϵ N∑c 1 2 2−ϵ g µ j j d q d Q − Ξ3(K, k, W) = i (τ τ ) 2 2−ϵ γd 1G3(K + Q, k + q) Nf (2π) (2π) j=1 ] 1 × γ − G (K + Q + W, k + q)γ − , (1.30) d 1 3 d 1 Q2 + c2q2

34 does not contain a 1/ϵ pole in the limit of c → 0. This subsequently leads to the lack of a pole in the two-loop vertex correction (the details of the computation are presented in Appendix A.1.3). For v, c ≠ 0 however, as described in Appendix A.3, the vertex correction picks up a small pole with a coefficient of O(c):

∫ ∫ dε (N 2 − 1)g2cµϵω1−ϵ 1 (1 − x)1/2(1 − v2) ⟨J J ⟩ (ω) ≈ − 3 c dx y y vert 3 2 2 2 3/2 2v 32π ϵ 0 (c + x(1 + v − c )) ( )− c2ε2 ϵ/2 ω2 + 3 (1.31) c2 + x(1 + v2 − c2)

However, this is subdominant to the self energy correction in Section 1.3.2, as the latter is finite in the limit c → 0; this is a consequence of a Ward identity discussed in Appendix A.3.

1.3.4 Renormalized conductivity σ(ω)

We can now add the leading free contribution in Eq. (1.22), the singular self-energy correction in Eq. (1.28) and the appropriate counter term to obtain the renormalization of the conductivity σ(ω) near the fixed point

∫ ∫ [ 1 2 dε3 NcNf −ϵ σyy(ω) ≈ (1 + v ) ω dx 1+ 2v 16π 0 ( )] (z − 1) (1 − x)1/2 ω2 c2ε2/µ2 ln + 3 . (1.32) π (c2 + x(1 + v2 − c2))1/2 µ2 c2 + x(1 + v2 − c2)

The interpretation of this central result requires some care in the limit of small v and c, and we consider various cases separately below. The important point here is that the argument of the logarithm is of order

2 2 2 (ω + (cε3) )/µ , and so the renormalization-group-improved perturbation expansion will lead to powers

2 2 2 of (ω + (cε3) )/µ , in contrast to the power of ω alone outside the square bracket. This difference arises because, at leading order, the singular contribution of the hot spot is the same as the rest of the Fermi surface, while at higher orders there is quasiparticle breakdown only close to the hot spot. Consequently, there is a modification in the nature of the ε3 integral, where we recall that ε3 measures distance away from the hot

35 spot along the Fermi surface.

First, let us assume the bare value of c is so small that the ε3 dependence of the argument of logarithm can be ignored; this was, effectively, the limit that was implicitly taken in by Abanov et al. [87]. This requires that cΛ < ω, where Λ is the momentum space cutoff. Then, we can easily perform the integral over x, and after re-exponentiating the logarithm in the ϵ expansion, we conclude that

∫ [ ] dε σ (ω) ∼ 3 ω−ϵ 1 + (z − 1) ln(ω/µ) yy 2v

∼ Λµ1−z ω−ϵ+(z−1). (1.33)

This is the answer expected from the hyperscaling violation case in Eq. (1.17) at this order in ϵ; Abanov et √ al. [87] found σ(ω) ∼ 1/ ω, which is consistent with Eq. (1.17) for their dynamic critical exponent z = 2.

Thus the ω dependence of σ violates hyperscaling as in Eq. (1.2) with dt = 1 for cΛ < ω < Λ, which can be within a universal regime only if the bare value of c is small enough, as claimed in Section 1.1.

Next, we consider the more generic case where the bare value of c is of order unity. Then, we can divide the integration over ε3 in Eq. (1.32) into two regimes. There is the far from the hot-spot regime where cε3 ≫ ω, and the close to the hot-spot regime of cε3 ≪ ω. The contribution of the close to the hot-spot regime is similar to that in Eq. (1.33), except that the upper bound on the integration over ε3 involves ω

∫ ω/c dε3 −ϵ+(z−1) σyy(ω) ∼ ω . (1.34) 0 2v

1/z Actually, by scaling, we expect the upper bound on the momentum integral over ε3/v to scale as ω at higher order in ϵ; using such an upper bound in Eq. (1.34) we obtain the generic hot-spot contribution

−ϵ+(z−1)+1/z σyy(ω) ∼ ω . (1.35)

36 To this order in ϵ, this is the scaling expected by the hyperscaling preserving scaling dimension in Eq. (1.16).

In comparison to the hyperscaling violating answer obtained in the direct v, c → 0 limit used for Eq. (1.33), the conductivity has acquired an extra factor of ω1/z. So we reach one of our main conclusions, that the hot- spot contribution to the conductivity generically obeys hyperscaling as in Eq. (1.1). We have not written out explicit factors of v and c in the final scaling forms, but these are ultimately only expected to yield powers of (ln(1/ω))−1, and so hyperscaling is only obeyed up to powers of ln(1/ω).

Finally, we also have to consider the contribution of the far from the hot spot regime cε3 ≪ ω. In this regime, the term inside the square brackets in Eq. (1.32) is ω-independent, and so we obtain an additional contribution σ ∼ ω−ϵ. This is just the additive Fermi liquid contribution of long-lived quasiparticles far from the hot spot.

1.4 T > 0 free energy

In order to study the finite temperature dynamics of this model, we need to compute the free energy density at T > 0. The free energy density has contributions from the free fermions, the free bosons, and a “self energy” correction due to their interactions. Following the lessons learned in the analysis of the optical conductivity in Section 1.3, we will perform the computation here only in the simpler limit of vanishing velocites v, c → 0, where we can replace the boson propagator by the momentum-independent form in

Eq. (1.26). However, as described in Section 1.3.4, we will assume that the low T hot spot contribution for the case of finite velocities can be estimated by limiting the range of the fermionic kx integral (along the

Fermi surface) to an upper limit ∼ T 1/z; here we have assumed the the upper cutoff is determined by T rather than ω for the optical conductivity in Section 1.3.4.

0 0 The free fermion, Ff , and free boson, Fb contributions to the free energy density, F , are obtained straightforwardly to leading order in ϵ (the prefactor of 4 in the fermion contribution comes from having 4 pairs of

37 hot spots)

∫ ∫ 1−ϵ [ ] dkx dkyd K¯ 2 ¯ 2 1/2 − 2 ¯ 2 1/2 F 0 = 4N N T ln (1 + e(ky+K ) /T )(1 + e (ky+K ) /T ) f c f 2π (2π)2−ϵ ∫ ( ) 3ζ(3) = dk N N T 3−ϵ , (1.36) x c f 2π2 where the infinite temperature-independent constant part was dropped. For the bosons

∫ ∫ 2 1−ϵ [ ] 2 d q d Q¯ 2 2 ¯2 1/2 π F 0 = (1 − N 2)T ln 1 − e−(c q +Q ) /T = (N 2 − 1)T 4−ϵ. (1.37) b c (2π)2 (2π)1−ϵ 90c2 c

Figure 1.4: The simplest interaction contribution to the free energy at O(g2).

The interaction contribution to the free energy at two-loop order is given by Fig. 1.4. It may be expressed as   N 2−1 ∫ 1 ∑c d2qd1−ϵQ¯ ∑ Π(q, T ) F = Tr  τ jτ j T , (1.38) fb 2 (2π)3−ϵ Q¯2 + c2|q|2 + ω2 j=1 ωq q where Π(q, T ) is the RPA polarization bubble given by

∫ ∑ 2 1−ϵ ¯ ∑ 2 d kd K Π(q, T ) = g T Tr [γ − G (k)γ − G (k + q)] . (1.39) (2π)3−ϵ d 1 n d 1 n n ωk

We separate out Π(q, T ) as

Π(q, T ) = (Π(q, T ) − Π(q, 0)) + Π(q, 0), (1.40)

38 and evaluate the finite temperature part setting v = 0 at the outset, taking g, c → 0 with g2/c finite and equal to its fixed point value λ∗w∗. The zero temperature part is evaluated with v ≠ 0 at the outset, and g, v → 0 with g2/v finite and equal to its fixed point value λ∗ . As described in Appendix A.2, this separates out the contributions that renormalize the free fermionic and bosonic contributions, with the finite temperature part of Π renormalizing the free fermionic contribution and the zero temperature part renormalizing the bosonic contribution. We then obtain (see Appendix A.2 for details), for the singular parts,

(1) (2) Ffb = Ffb + Ffb , ∫ ( ) g2 3ζ(3) F (1) = dk (N 2 − 1)T 3−2ϵ , fb x c c 16π3ϵ g2π F (2) = (N 2 − 1)T 4−2ϵ . (1.41) fb c 360vc2ϵ

We have set the momentum renormalization scale µ = 1 in the present section. We thus get, after plugging in the fixed point values,

∫ ( ) ∫ 3ζ(3) T −ϵ 3ζ(3) F = F 0 + F (1) = dk N N T 3−ϵ 1 + (z − 1) → dk N N T 3−ϵ−(z−1), f f fb x 2π2 c f ϵ x 2π2 c f (1.42) where the pure 1/ϵ pole is cancelled by the usual addition of a counter term. Similarly,

( ) π2 z − 1 π2 F = F 0 + F (2) = (N 2 − 1)T 4−ϵ 1 + 2 T −ϵ → (N 2 − 1)T 4−ϵ−2(z−1). (1.43) b b fb 90c2 c ϵ 90c2 c

2−ϵ+2/z We now observe that the bosonic term Fb is compatible with the behavior ∼ T expected from the hyperscaling preserving scaling dimension in Eq. (1.14); the agreement holds to first order in ϵ after recalling that z − 1 is O(ϵ) from Eq. (1.12). For the fermionic contribution, as in Section 1.3.4 the behavior depends upon the fate of the kx integral. As noted at the beginning of the present section, for the low T behavior we

1/z 3−ϵ−(z−1)+1/z should impose an upper cutoff on the integral of order T ; then Ff ∼ T which also agrees

39 with ∼ T 2−ϵ+2/z to first order in ϵ. Thus both the bosonic and fermionic contributions to the free energy obey hyperscaling, and the behavior in Eq. (1.1), up to logarithms.

As was the case in Section 1.3, for very small bare velocity c, and for cΛ < T < Λ, there is a regime of hyperscaling violation when the kx integral is replaced by Λ, and behavior is as in Eq. (1.2). Note that we are using units in which the velocity vF in Eq. (1.7) has been set equal to unity; so the full condition for this intermediate regime is cΛ < T < vF Λ.

1.5 Quantum Boltzmann Equation

We now compute the hot spot conductivity σQ appearing in Eq. (1.4) in d = 2 using a quantum Boltzmann equation approach [102–105]. We use the Keldysh formalism at one-loop order to derive quantum kinetic equations for the fermions and bosons in the presence of an applied electric field, and then solve these equations in linear response to obtain the contribution of the fermions near the hot spots to the DC conductivity.

Note that, unlike the previous sections, we are not performing a systematic ϵ expansion here, but working directly in d = 2 to minimize technical complexity.

As in Section 1.4, we will restrict our analysis to the case of vanishing v and c, when the Fermi surfaces are nested, and manipulations similar to Eq. (1.26) can be applied. With finite v and c, as argued in Sections 1.3.4 and 1.4, we can estimate the low T hot spot conductivity by limiting the kx integral along the Fermi surface by an upper bound of order T 1/z.

1.5.1 Keldysh framework

We begin by expressing the action in Eq. (1.6) on the closed time Keldysh contour [104, 105]. Denoting with subscripts + the forward part of the contour and with subscripts − the backward part of the contour, we

40 obtain for the free part of the action

Sψψ¯ ∫ ∫ [ ∞ 2 ∑4 ∑ ∑ d p ¯(m) − m (m) = dt 2 ψℓ,σ,+(t, p)(i∂t eℓ (p)) ψℓ,σ,+(t, p) −∞ (2π) ↑ ↓ ℓ=1 m= σ= , ] − ¯(m) − m (m) ψℓ,σ,−(t, p)(i∂t eℓ (p)) ψℓ,σ,−(t, p) , ∫ ∫ [ ] ∞ 2 ( ) ( ) 1 d q 2 2 2 2 − · − − − − − · − − − Sϕϕ = dt 2 ϕ+(t, q) ∂t ωq ϕ+(t, q) ϕ (t, q) ∂t ωq ϕ (t, q) , 2 −∞ (2π)

(1.44)

with ωq = c|q|. The interacting part is given by

[ ∫ ∞ ∫ ∑4 ∑ (+) (−) − 2 · ¯ ′ − Sϕψψ¯ = g dt d r ϕ+(t, r) ψℓ,σ,+(t, r)τσ,σ ψℓ,σ′,+(t, r) −∞ ′ ↑ ↓ ℓ=1 σ,σ = , ] (+) (−) − · ¯ ′ ϕ−(t, r) ψℓ,σ,−(t, x)τσ,σ (t, r)ψℓ,σ′,−(t, r) + h.c. , (1.45)

We now perform the standard bosonic and fermionic Keldysh rotations: For the real bosons we use

ϕ+ = ϕc + ϕq,

ϕ− = ϕc − ϕq (1.46)

and for the Grassmannian fermions we have

1 1 ψ(m) = √ (ψ(m) + ψ(m) ), ψ¯(m) = √ (ψ¯(m) + ψ¯(m) ), ℓ,σ,+ 2 ℓ,σ,1 ℓ,σ,2 ℓ,σ,+ 2 ℓ,σ,1 ℓ,σ,2 (m) 1 (m) (m) ¯(m) 1 ¯(m) ¯(m) ψ − = √ (ψ − ψ ), ψ − = √ (ψ − ψ ). (1.47) ℓ,σ, 2 ℓ,σ,1 ℓ,σ,2 ℓ,σ, 2 ℓ,σ,2 ℓ,σ,1

41 Hence, we get for the free fermion part of the lagrangian

    [ ] −1 ∑4 ∑ ∑ ( )  GRℓm δK   ψ(m) (t, p)  L ¯(m) ¯(m)  0 f   ℓ,σ,1  ψψ¯ = ψℓ,σ,1(t, p) ψℓ,σ,2(t, p)     , [ ]− ℓ=1 m= σ=↑,↓ Aℓm 1 (m) 0 G0 ψℓ,σ,2(t, p)

(1.48)

K where the infinitesimal δf ensures convergence. Inverting this matrix, we obtain the bare fermion Green’s function matrix

   GRℓm GKℓm  bℓm  0 0  G0 =   . (1.49) Aℓm 0 G0

The bare retarded (R) and advanced (A) fermion Green’s functions thus are

1 GRℓm(ω, p) = , 0 − m ω + i0+ eℓ (p) 1 GAℓm(ω, p) = . (1.50) 0 − − m ω i0+ eℓ (p)

For the free boson part of the lagrangian we have

    [ ] A −1 1  0 D0   ϕc(ω, q)  L = (ϕ (−ω, −q) ϕ (−ω, −q))     , (1.51) ϕϕ 2 c q  [ ]    R −1 K D0 δb ϕq(ω, q)

K where the infinitesimal δb again ensures convergence. After performing the matrix inverse,

   DK DR  b  0 0  D0 =   (1.52) A D0 0

42 and the retarded and advanced boson Greens’ functions hence are

1 1 DR(ω, q) = , 0 2 − 2 2 (ω + i0+) ωq 1 1 DA(ω, q) = . (1.53) 0 − 2 − 2 2 (ω i0+) ωq

The interaction between fermions at the (ℓ, +) and (ℓ, −) hot spots and the boson takes the following form:

    − 4 ( ) ( ) ∑ ∑  ϕc(t, r) · τσσ′ ϕq(t, r) · τσσ′   ψ ′ (t, r)  L − ¯(+) ¯(+)    ℓ,σ ,1  ϕψψ¯ = g ψℓ,σ,1(t, r) ψℓ,σ,2(t, r)     ′ (−) ℓ=1 σ,σ =↑,↓ · ′ · ′ ϕq(t, r) τσσ ϕc(t, r) τσσ ψℓ,σ′,2(t, r)

+ h.c. (1.54)

This gives rise to the Feynman rules summarized graphically in Fig. 1.5.

We adopt the shorthand convention of x = (t, r) and q = (ω, q) to combine spatial and temporal components. We have the relations

Kℓm Rℓm Aℓm G = G ◦ Ff − Ff ◦ G ,

K R A D = D ◦ Fb − Fb ◦ D , (1.55)

∫ ′ ′ where C = A ◦ B implies C(x, x ) = dx1A(x, x1)B(x1, x ) and Ff,b are respectively the fermionic and bosonic distribution functions. The Dyson equations for the matrix fermion and boson Green’s functions are

bℓm −1 − bℓm ◦ bℓm ([G0 ] Σf ) G = 1,

b−1 − b ◦ b (D0 Σb) D = 1. (1.56)

The self energy matrices Σb have the same form as the inverse Green’s function matrices in Eq. (1.49) and

43 Figure 1.5: Feynman rules and one-loop graphs for the self energies of the fermions and bosons in the Keldysh formalism: (a) Fermion propagators. (b) Boson propagators. (c) Yukawa vertices. (d) Self energies. Here x = (t, r), hot spot indices (ℓ, ) are suppressed and the external legs on the self energy diagrams are amputated. The external momentum and frequency on the self energy diagrams is on shell. The 2-1 and q-q propagators are zero and hence are omitted.

Eq. (1.51), and hence the different components of the self energies are given by the graphs in Fig. 1.5 at the

′ ′ one loop level. Defining central and relative coordinates xc = (x + x )/2 and xr = (x − x )/2, we can

′ convert the two point functions G, D, F and Σ which are of the form A(x, x ) = A(xc + xr/2, xc − xr/2) into a momentum representation via the Wigner transform

∫ ( ) − x x A(x , p) = dx e ipxr A x + r , x − r . (1.57) c r c 2 c 2

Since we have spatial translational invariance in the linear response limit a of weak applied electric field E,

44 we can further simplify A(xc, p) → A(t, p). We will also always consider external particles to be on shell in the subsequent computations of the collision integrals. We define an alternate parameterization ff,b of the distribution functions Ff,b

Ff (t, p, ω) = 1 − 2ff (t, p, ω),

Fb(t, q, ω) = 1 + 2fb(t, q, ω). (1.58)

In thermal equlibrium in the absence of any applied electric fields, we have ff,b(t, k, ω) = nf,b(ω), where

ω/T nf,b(ω) = 1/(1 e ) are the thermal Fermi and Bose functions respectively [105].

1.5.2 Kinetic equations for fermions and bosons

There are two coupled quantum kinetic equations [105, 107], one for the electrically charged fermions,

( ) ∂ ∂ ℓ coll + E · F (t, p) = I [F ,F ](t, p), (1.59) ∂t ∂p f fℓ f b

ℓ with the on shell fermion distribution function Ff (t, p) = Ff (t, p, eℓ (p)); and one for the neutral bosons,

∂ F (t, q, ω ) = Icoll[F ,F ](t, q), (1.60) ∂t b q b f b

The fermion electric charge is set to 1 for simplicity. The two collision integrals have the general form [105]

[ ] coll Kℓ ℓ ℓ Rℓ ℓ Ifℓ[Ff ,Fb](t, p) = iΣf (t, p, e (p)) + 2Ff (t, p)Im Σf (t, p, e (p)) , [ ] coll K R Ib [Ff ,Fb](t, q) = iΣb (t, q, ωq) + 2Fb(t, q, ωq)Im Σb (t, q, ωq) . (1.61)

45 R,Kℓ At the one-loop level, Σf,b are given by the graphs in Fig. 1.5. The self energies and collision integrals are computed in Appendix A.4. We obtain

coll Ifℓ[ff , fb](t, p) = ∫ ( 2 [ 2 d q 1 − ∓ − − ℓ − ℓ∓ − − ℓ∓ − = 3g δ(eℓ (p) eℓ (p q) ωq) ff (t, p)(1 ff (t, p q)) ff (t, p q)fb(t, q, ωq)+ 2π ωq ] [ ℓ − − ∓ − ℓ − ℓ− − − + ff (t, p)fb(t, q, ωq) δ(eℓ (p) eℓ (p q) + ωq) ff (t, p)(1 ff (t, p q)) ) ] − ℓ∓ − − ℓ − ff (t, p q)fb(t, q, ωq) + ff (t, p)fb(t, q, ωq) , (1.62)

and

∫ ∑ 2 [ ( ) ( d k − − Icoll[f , f ](t, q) = 4g2 δ e (k) + ω − e+(k + q) f ℓ+(t, k + q)(1 − f ℓ (t, k))+ b f b 2π ℓ q ℓ f f ℓ ) ] ℓ+ − ℓ− ↔ − + ff (t, k + q)fb(t, q, ωq) ff (t, k)fb(t, q, ωq) + (+ ) , (1.63)

where we have expressed the collision integrals in the alternate parameterization (1.58) of the distribution

ℓ functions Ff,b and ff (t, p) = ff (t, p, eℓ (p)).

1.5.3 Ansatz and solution for conductivity

If we set the collision integrals to zero, the distribution function for the neutral bosons unaffected by the applied electric field remains fixed at its equilibrium value. For the Fermions, we have

( ) ∂ ∂ + E · f ℓ (t, p) = 0. (1.64) ∂t ∂p f

46 To solve this in linear response, we switch from the time to the frequency domain and parameterize the

ℓ deviation of ff (ω, k) from its equilibrium value by [107, 108]

ℓ · · · · − · ff (ω, p) = 2πδ(ω)nf (vℓ p) + vℓ E(ω)φ(vℓ p, ω)nf (vℓ p)(1 nf (vℓ p)). (1.65)

Inserting this parameterization into Eq. (1.64), we obtain the collisionless φ function in linear response

1/T φ (v · p, ω) = . (1.66) nc ℓ −iω + 0+

We have the electrical current density:

∫ ∫ ∑4 ∑ 2 ∑4 ∑ 2 d p † d p J(ω) = 2 vm⟨ψm ψm⟩(ω, p) = 2 vmf ℓm(ω, p), (1.67) (2π)2 ℓ ℓ ℓ (2π)2 ℓ f ℓ=1 m= ℓ=1 m= and hence obtain the linear response conductivity

∫ ∑4 ∑ 2 δJi(ω) d p m · m · · · − · σij(ω) = = 2 2 (vℓ ^ei)(vℓ ^ej)φ(vℓ p, ω)nf (vℓ p)(1 nf (vℓ p))). (1.68) δEj(ω) (2π) ℓ=1 m=

It is easily seen that if the φnc is used in the above expression, we obtain a temperature independent collisionless conductivity that has a delta function in ω in its real part. In fact, for the Sur-Lee embedding in higher dimensions, we have for the collisionless conductivity

∫ nc nc ∝ 1−ϵ Re[σxx(ω)] = Re[σyy(ω)] dkx T δ(ω), (1.69)

which is derived in Appendix A.4; assuming the kx integral yields a factor of the cutoff Λ, this yields a conductivity with the hyperscaling violating scaling dimension in Eq. (1.17), as expected for free fermions.

Once collisions of the fermions with the bosons are included, these delta functions are broadened, and the

47 kx integral has to be performed with more care, as in Section 1.3.4.

Returning to d = 2, we find that, to linear order in E, the bosons still remain in equilibrium and their distribution function is hence given by the thermal Bose function fb(t, k, ωk) = nb(ωk) if the parameterization

Eq. (1.65) is used (See Appendix A.4 for a derivation of this fact). Intuitively, this is because the linearly dispersing hot spot model exhibits particle-hole symmetry, making the charge carrying modes excited by the applied electric field particle-hole pairs with the particle and hole moving in opposite directions. The bosons then do not absorb any momentum that they have to dissipate when the particle-hole pairs recombine and hence remain in equilibrium. This behavior is also present for quantum critical transport in graphene [107].

We now consider the system to be at the fixed point discussed previously, in the spirit of [102]. We take the applied electric field to be in the y direction (E = Ey^ey); since v → 0 at the fixed point, only the ℓ = 1 and ℓ = 3 pairs of hot spots contribute significant response in this case. (For the electric field in the x direction we obtain the same response with the ℓ = 4 and ℓ = 2 hot spot pairs respectively). We insert the

ℓ ff functions parameterized by Eq. (1.65) and the thermal Bose function for fb into the frequency domain version of the fermion kinetic equation Eq. (1.59), and linearize in Ey to get the following integral equation for φ in the v → 0 limit (considering the ℓ = 1 pair of hot spots)

( ∫ [ 1 3g2 d2q − δ(2p − q − c|q|) φ(p , ω)n (p ) (1 − n (p )) (1 − n (q − p ) + n (c|q|)) + 2 2π c|q| y y y f y f y f y y b ] + φ(qy − py, ω)nf (qy − py) (1 − nf (qy − py)) (nf (py) + nb(c|q|)) − [ − δ(2py − qy + c|q|) φ(py, ω)nf (py) (1 − nf (py)) (1 − nf (qy − py) + nb(−c|q|)) + ) ] + φ(qy − py, ω)nf (qy − py) (1 − nf (qy − py)) (nf (py) + nb(−c|q|))

1 = (−iω + 0+)φ(p , ω)n (p ) (1 − n (p )) − n (p ) (1 − n (p )) . y f y f y T f y f y

(1.70)

48 In the collision term, the boson momentum parallel to the Fermi surface, qx, is limited by the Bose function to a value of order T/c. Integrating out qx, we obtain

[ ] 2 3g C1[φ, py] + 1 C0(py)φ(py, ω) + = (−iω + 0 )φ(py, ω) − . (1.71) 2πc nf (py) (1 − nf (py)) T where,

∫ [ ( ) ] ∞ − 2 − 2 2 1 Θ (qy 2py) c qy C0(py) = dqy sgn(qy − 2py) √ (1 − nf (qy − py) + nb(2py − qy)) , 2 −∞ − 2 − 2 2 (qy 2py) c qy ∫ [ ( ) ∞ − 2 − 2 2 1 Θ (qy 2py) c qy C1[φ, py] = dqy sgn(qy − 2py) √ φ(qy − py, ω)nf (qy − py)× 2 −∞ − 2 − 2 2 (qy 2py) c qy ]

× (1 − nf (qy − py)) (nf (py) + nb(2py − qy)) . (1.72)

This equation may be solved iteratively by choosing the trial solution

−1/T φ1(py, ω) = (1.73) 3g2 − + 2πc C0(py) + iω 0 and iterating

( ) 2 2 3g + 1 3g C1[φj, py] C0(py) + iω − 0 φj+1(py, ω) = − − , (1.74) 2πc T 2πc nf (py) (1 − nf (py))

for j > 1 (Note that φ1 in Eq. (1.73) may be derived from inserting φ0 = 0 into Eq. (1.74)). The integral for C1 may be evaluated numerically by sampling φ at a discrete set of points and then constructing an interpolating function through these points. The exponential decay of nf,b at large values of their arguments ensures convergence of the integrals and supresses errors arising from the extrapolation of the interpolating function to large arguments. The trial solution φ1 is actually fairly accurate, and this algorithm converges in

49 a small number of iterations.

In the limit of c → 0, which also occurs at the fixed point, we see that C0 ∼ 1/c because the singularity in nb(2py − qy) as 2py − qy → 0 is cut off by c in the Θ function in Eq. (1.71); this also occurs in the integral for C1 in (1.72). Hence in this limit, we have for ω → 0,

( ) c2 p φ (p , 0) ≈ χ y (1.75) 1 y g2T T for some function χ. It can be seen from Eq. (1.74), and also established numerically, that given the above form for φ1, all φj>1 will also be of the same form. Numerically, we find that χ is an even function and

χ(0) = 0. Since χ is an even function, it is easy to show that the (1, −) hot spot contributes the same value to the conductivity as the (1, +) hot spot as it is related by the transformation of py → −py in the above computation. Similarly, the (3, ) hot spots also produce identical contributions equal to those from the

(1, ) hot spots. Hence, using Eq. (1.68),

∫ ( ) ∫ c2 d2p p c2 σ (0) ≈ 8 χ y n (p )(1 − n (p ))) ∼ dk . (1.76) yy g2T (2π)2 T f y f y x g2

In the last step we have changed the fermion momentum notation from px to kx for compatibility with earlier discussions. It is also easily seen that σxx = σyy if we repeat the above analysis for the ℓ = 2 and ℓ = 4 hot spots instead, and that σxy = σyx = 0. So Eq. (1.76) is the estimate by the Boltzmann equation of the value of the conductivity σQ in Eq. (1.4).

We now need to determine the T dependence implied by Eq. (1.76) using scaling ideas. Under the renormalization group flow, we expect that the coupling λ = g2/v flows to a fixed point value. While this fixed point can be determined precisely under an ϵ expansion, we are only able to make an estimate in the present computation carried out directly in d = 2, where g2/v is a dimensionful quantity of order µϵ. The natural scale for the momentum µ is set by the temperature, and so µ ∼ T 1/z. So in d = 2, we can expect that

50 g2/v ∼ T 1/z. Ignoring the logarithmic factors, we therefore have the estimate

∫ −1/z σQ ∼ dkx T . (1.77)

1/z Finally, as in Sections 1.3 and 1.4, we bound the kx integral by T to conclude that σQ ∼ constant, as

−1/z claimed in Section 1.1. And also as in previous sections, for a small bare c, we will have σQ ∼ ΛT in the intermediate T regime cΛ < T < Λ (and as noted at the end of Section 1.4, after restoring units, this condition is cΛ < T < vF Λ).

1.6 Discussion

We have computed the critical conductivity and free energy at the onset of spin density wave order in metals in d = 2 using the ϵ expansion introduced by Sur and Lee [89]. The advantage of this method is that the ϵ expansion appears to be valid systematically order-by-order in ϵ, and there is no breakdown in the renormalization group flows. The ϵ expansion exhibits a logarithmic flow of the velocity v to zero at large length scales, and a dynamic nesting of the Fermi surfaces. We found that hyperscaling was obeyed, with the hot spot contributions scaling as in Eq. (1.1).

It is interesting to compare these results with a previous two-loop, large N renormalization group analysis of the spin-density wave critical point in Ref. [86], which also found a logarithmic flow of v to zero at low energies. However, it was also found that the large N expansion broke down at sufficiently large scales.

The same large N framework was used to compute the optical conductivity by Hartnoll et al. [84], and it was found that hot-spot contribution was σ(ω) ∼ constant in the limit of vanishing v, as expected under hyperscaling in d = 2.

This similarity between the large N and ϵ expansion indicates that the terminology ‘quasi-local’ for the latter expansion [89] should be used with some care, and we have avoided it here. The basic scaling prop-

51 erties are similar to those of a standard, spatially-isotropic, critical point obeying hyperscaling with a finite dynamic critical exponent z given by Eq. (1.12). The deviations from strong scaling arise only in logarithmic corrections, which are linked, ultimately, to the asymptotic nesting of the Fermi surfaces [85, 86] in

Fig. 1.1b.

We also carried out computations for the free energy density at non-zero T using the ϵ expansion. Again our results were in excellent accord with hyperscaling expectations. Both the fermionic excitations at the hot spot and the collective bosonic fluctuations scaled with the same power of T , as shown in Section 1.4.

There was, however, for the somewhat unnatural case of a sufficiently small bare boson velocity, an intermediate energy regime where hyperscaling was violated. This was discussed in Section 1.3 for the optical conductivity, and in Section 1.4 for the free energy. The optical conductivity results of Refs. [87, 88] are similar to this hyperscaling violating regime, and our analysis indicates that their results do not apply when the bare boson velocity is not small.

Finally, in Section 1.5, we addressed the question of the DC conductivity. Because of the constraints of total momentum conservation, such a computation must be carried out [60, 62] in the context of the expression in Eq. (1.4), which separates a quantum critical conductivity σQ from that associated with ‘drag’

0 from the conserved momentum. We estimated σQ in Section 1.5 and found a result that scaled as T , up to logarithms. Thus, the σQ contribution to Eq. (1.5), in the theory of the spin density wave critical point, is likely not the mechanism of the strange metal linear resistivity.

The momentum-drag term in Eq. (1.5) was considered in a previous work [66] for the spin density wave critical point: there we found that quenched disorder which changes the local critical coupling did lead to a linear-in-T resistivity. This conclusion is not modified by the considerations of the present chapter.

52 Clarity is momentum that has no resistance in it.

Esther Hicks

2 DC resistivity at the onset of spin density wave

order in two-dimensional metals

2.1 Introduction

A wide variety of experiments on correlated electron compounds call for an understanding of the transport properties of quasi-two-dimensional metals near the onset of spin density wave (SDW) order [8, 22, 109, 110].

Nevertheless, despite several decades of intense theoretical study [24, 25, 84–87, 89, 90, 106, 111–116], the basic experimental phenomenology is not understood. A common feature of numerous experimental studies

[109, 117] is a non-Fermi liquid behavior of the resistivity, which varies roughly linearly with temperature

53 at low T , and more rapidly at higher T .

The conventional theoretical picture of transport [90, 112] is that the non-Fermi liquid behavior of the electronic excitations is limited to the vicinity of a finite number of “hot spots” on the Fermi surface: these are special pairs of points on the Fermi surface which are separated from each other by K, the ordering wavevector of the SDW. The remaining Fermi surface is expected to be ‘cold’, with sharp electron-like quasiparticles, and these cold quasiparticles short-circuit the electrical transport, leading to Fermi liquid behavior in the DC resistivity.

Recent theoretical works [84, 116, 118] have called aspects of this picture into question, and argued that the cold portions of the Fermi surface are at least ‘lukewarm’. Composite operators in the quantum-critical theory can lead to strong scattering of fermionic quasiparticles at all points on the Fermi surface. Perturbatively, the deviation from Fermi liquid behavior is strongest at the hot spots, but the quantum critical theory flows to strong coupling [86], and so we can expect significant deviation from Fermi liquid physics all around the

Fermi surface.

In the context of the DC resistivity, an important observation is that all of these deviations from Fermi liquid behavior arise from long-wavelength processes in an effective field theory for the quantum critical point. Consequently, they are associated with the conservation of an appropriate momentum-like variable, and one may wonder how effective they are in relaxing the total electrical current of the non-Fermi liquid state. For commensurate SDW with 2K equal to a reciprocal lattice vector, it may appear that, because the interactions allow for umklapp, conservation of total momentum is not an important constraint. However, as we will argue in more detail below, once we have re-expressed the theory in terms of the collective modes of the effective field theory, a suitably defined momentum is conserved and its consequences have to be carefully tracked. It is worthwhile to note here that a similar phenomenon also appeared in the theory of transport in the Luttinger liquid in one spatial dimension by Rosch and Andrei [119], where a single umklapp term was not sufficient to obtain a non-zero resistivity.

54 The present chapter will address the question of the T dependence of the DC resistivity at the SDW quantum critical point using methods which represent a significant departure from the perspective of previous studies [87, 90, 112]. We shall employ methods similar to those used recently [62] for the Ising-nematic quantum critical point, which was inspired by analyses of transport in holographic models of metallic states

[60, 61, 64, 97, 100, 120–127], and by Boltzmann equation studies [128, 129]. Related methods have also been used for transport in non-Fermi liquids in one spatial dimension [119, 130–132].

The central assumption underlying these approaches is that the momentum-conserving interactions responsible for the non-Fermi liquid physics are also the fastest processes leading to local thermal equilibration. We will assume here that excitations near both the hot and lukewarm portions of the Fermi surface are susceptible to these fast processes, and are able to exchange momentum rapidly with each other. Then we have to look towards extraneous perturbations to relax the total momentum, and allow for a non-zero

DC resistivity. These perturbations can arise from impurities, from additional umklapp processes beyond those implicitly contained in the field theory, or from coupling to a phonon bath. Here we will focus on the impurity case exclusively, and leave the phonon contribution for future study. The umklapp contribution can also be treated by the present methods [61, 97], and, in the approximation where cold fermions are present, yield a conventional T 2 resistivity.

For our subsequent discussion, it is useful to introduce a specific model for the SDW quantum critical point. We find it convenient to work with a two-band model, similar to that used recently for a sign-problem- free quantum Monte Carlo study [115]. Closely related models have been used for a microscopic description of the pnictide superconductors [133–136]. As was argued in Ref. [115], we expect our conclusions to also apply to SDW transitions in single band models because the single and two band models have essentially the same Fermi surface structure in the vicinities of all hot spots. Our model begins with two species of

e e fermions, ψa, ψb which reside in pockets centered at (0, 0) and (π, π) in the square lattice Brillouin zone, as shown in Fig. 2.1(a). We take the SDW ordering wavevector K = (π, π). Then, we move the pocket

55 Figure 2.1: (a) The two pockets of fermions separated by the SDW ordering wavevector K = (π, π). (b) The resulting pair of Fermi surfaces after shifting the pocket centered at (π, π) to (0, 0) intersect at 4 hot spots as shown.

e e iK·r centered at (π, π) and move it to (0, 0) by introducing fermions ψa(r) = ψa(r) and ψb(r) = ψb(r)e : the

Fermi surfaces for the ψa, ψb fermions are shown in Fig. 2.1(b). The advantage of the latter representation is that the coupling of the fermions to the SDW order parameter ϕ is now local and r independent. So we can now write down a continuum Lagrangian for the SDW quantum critical point in imaginary-time (t → −iτ)

   ( )  ξa 0  1 ϵ u 3 2 L = ψ† ∂ − µ +   ψ + ∇ϕ · ∇ϕ + (∂ ϕ )(∂ ϕ ) + ϕ ϕ − + λψ†ϕ Γ ψ.  τ 0   2 µ µ 2 τ µ τ µ 6 µ µ g µ µ 0 ξb (2.1)

We have two species of spin 1/2 fermions (a, b) with chemical potential µ0 in two spatial dimensions coupled ( ) to a SO(3) vector boson order parameter ϕ . We have ψ = ψa where ψ are two-component spinors. µ ψb a,b

56 ( ) The matrices Γ = 0 σµ with σ as the Pauli matrices acting on the spin indices only. The dispersions µ σµ 0 µ of the fermions are 2 2 2 2 ∂x ∂y ∂x ∂y ξa = − − + . . . , ξb = − − + ... (2.2) 2m1 2m2 2m2 2m1

This produces two Fermi surfaces intersecting at four hot-spots (Fig. 2.1(b)). Higher-order derivatives in

Eq. (2.2) are allowed provided additional Fermi surfaces do not appear at larger momenta. At the critical point, we choose the value of g so that the coefficient of ϕµϕµ vanishes. We can now take the lower energy theory in the vicinities of the 4 hot spots in Fig. 2.1(b), and obtain a model identical to that studied in numerous earlier works[84–87, 89, 113]. In particular, all of the computations on the optical conductivity in Ref. [84] apply essentially unchanged to the present continuum model L.

Now a key observation is that the resistivity of the model L is identically zero, ρ(T ) = 0, at all T .

This follows immediately from the translational invariance of L and the existence of an exactly conserved momentum which we will specify explicitly in Section 2.2. So we must include additional perturbations to

L will break the continuous translational symmetry to obtain a non-zero resistivity. One such perturbation is a random potential, which can scatter fermions at all momenta (including a → b processes that actually change momenta by K). It is given by

† † LV = V1(r)ψ (r)ψ(r) + V2(r)ψ (r)Γ0ψ(r), (2.3)

0 1 where Γ0 = ( 1 0 ). The other is a random-mass term for the bosonic field:

Lm = m(r)ϕµ(r)ϕµ(r), (2.4)

which corresponds to a local random shift in the position of the SDW quantum critical point. The random

57 terms are chosen to satisfy the following upon averaging over all realizations:

⟨⟨ ⟩⟩ ⟨⟨ ′ ⟩⟩ 2 2 − ′ V1,2(r) = 0 ; V1,2(r)V1,2(r ) = V0 δ (r r ),

⟨⟨ ⟩⟩ ⟨⟨ ′ ⟩⟩ 2 2 − ′ m(r) = 0 ; m(r)m(r ) = m0δ (r r ). (2.5)

The random-mass is expected to be a relevant perturbation to the SDW quantum critical point of L, and we will see that it also has a strong influence on the DC transport.

One of our main results is the following low T contribution of the random-mass perturbation to the resistivity, in general accord with the scaling arguments in Refs. [62] and [100]:

∼ 2 2(1+∆−z)/z ρm(T ) m0 T , (2.6)

where z is the dynamic scaling exponent, and ∆ is identified here with the dimension of the ϕ2 operator. In general, the latter is related to the correlation length exponent, ν, via

1 ∆ = d + z − . (2.7) ν

Note that this contribution arises from the disorder coupling to the bosonic critical modes of the quantum critical theory, and so is driven primarily by long-wavelength disorder. In the conventional Hertz-like limit of the SDW critical point [24, 25] we have d = 2, z = 2, and ν = 1/2, in which case Eqs. (2.6,2.7) yield

ρm(T ) ∼ T , one of our main results. Our explicit computation also finds logarithmic corrections. At higher temperature, we can envisage a crossover from the z = 2 Hertz regime, to a z = 1 Wilson-Fisher regime

[62, 137–141]: here for d = 2, z = 1, and[142] ν ≈ 0.70, Eqs. (2.6,2.7) yield ρ(T ) ∼ T 3.14. We note that a different discussion of the influence of disorder on the bosonic modes appeared recently. [143]

We also compute the contribution of the random potential terms in LV to the resistivity. Here the dominant

58 contribution is from the scattering of fermions between hot spots, and so this requires disorder at the short- wavelengths corresponding to the separation between the hot spots. These lead, as expected, to a leading term which is a constant as T → 0. However, we find that the leading vertex correction has an additional contribution from scattering of fermions between hot spots which varies linearly with T (up to logarithmic corrections) at low T . So we have

∼ 2 ρV (T ) V0 (1 + c T ), (2.8) for some constant c. Interestingly, we find that the vertex correction contribution is linear in T even in the z = 1 regime.

A notable point above is that the residual resistivity arises solely from the fermionic contribution associated with LV , and requires short-wavelength disorder. In contrast, the linear resistivity of ρm(T ) arises from the bosonic order parameter fluctuations coupling to long-wavelength disorder. Thus there is no direct correlation between the magnitudes of the residual resistivity and the co-efficient of the linear resistivity.

All of the considerations of this chapter also apply to other density wave transitions in two-dimensional metals, including the onset of charge density wave order. We only require that the order parameter have a non-zero wavevector which connects two generic points on the Fermi surface, and assume that the quantum critical theory is strongly coupled. No other feature of the spin density wave order is used in our analysis, and we focus on it mainly due to its experimental importance.

The body of this chapter describes our computation of the DC resistivity of L + LV + Lm. The outline is as follows: In Section 2.2 we discuss the continuous symmetries and derive the conserved currents of our model. In Section 2.3 we discuss the application of the memory matrix formalism to the calculation of the

DC resistivity. In Section 2.4 we obtain the contributions of the random mass term and random potential terms to the DC resistivity using the memory matrix formalism. We present details of the computations of all required quantities in the appendices.

59 2.2 Symmetries and Noether currents

The Lagrangian L is invariant under the following symmetries (translation, global U(1) symmetry and global

SU(2) spin rotation symmetry):

x → x + a, τ → τ + a0,

ψ → eiαψ, ( ) i θj σj iθj sj ψ → e 2 ψ, ϕµ → e ϕν. (2.9) ν

where sj are the generators of SO(3).

The above mentioned symmetries produce various conserved currents which may be derived using the standard Noether procedure; Translational symmetry produces

( ) ∑ ∂L ∂L T = ∂ ζ − ∂ ∂ ζ − δ L, (2.10) ab ∂(∂ ζ ) b n a ∂(∂2ζ ) b n ab n a n c n

where a, b are spatial indices and ζn are all the fields involved (in this case ψ and ϕµ). Time translational invariance giving the Hamiltonian density H (T00) and momentum density P (T0i) (with πµ =

2 −i∂L/(∂(∂τ ϕµ)) = −iϵ∂τ ϕµ, and the equal time commutation relation [ϕµ(x), πν(y)] = δ (x − y)δµν):

† H(ψ, ϕµ, πµ) = −ψ ∂τ ψ − ϵ(∂τ ϕµ)(∂τ ϕµ) + L(ψ, ∂τ ψ, ϕµ, ∂τ ϕµ), i P = − (ψ†∇ψ − ∇ψ†ψ) + π ∇ϕ . (2.11) 2 µ µ

∫ 2 Since ∂µTµν = 0, ∂τ d x(H, P) = boundary terms = 0.

60 The U(1) symmetry produces

( ) ∑ ∂L ∂L δζ j = − ∂ n , (2.12) µ ∂(∂ ζ ) µ ∂(∂2ζ ) δα n µ n µ n which gives the current density J :

( ) J i 1 † − † 1 † − † x = (∂xψaψa ψa∂xψa) + (∂xψbψb ψb∂xψb) , 4 ( m1 m2 ) J i 1 † − † 1 † − † y = (∂xψaψa ψa∂xψa) + (∂xψbψb ψb∂xψb) . (2.13) 4 m2 m1

The SU(2) symmetry produces spin currents but they cannot be used with the memory matrix approach, as explained below.

2.3 Memory Matrix Approach

The above theory does not possess well defined quasiparticles in two dimensions near the quantum critical point due to the strong (non-irrelevant) coupling λ, and hence it is not possible to correctly calculate transport properties like resistivity using traditional methods, as these involve doing perturbation theory in the coupling. ∫ However, the presence of a conserved total momentum P = d2xP, which will slowly relax if perturbations such as a weak disordered potential are applied, allows certain transport properties such as the DC resistivity to be correctly calculated using the memory matrix formalism [62, 97, 144].

In this formalism, the conductivity tensor σij may be expressed as [97, 144]

( )

i σij(ω) = Ji Jj , (2.14) ω − L

∫ with J = d2xJ , the Liouville super operator L acting as A(t) = eiHtAe−iHt = eiLtA(0), and the inner

61 ∫ | β ⟨ † ⟩ ⟨ ⟩ product of operators (A B) = 0 dτ A (τ)B(0) , with .,. denoting the connected correlation function.

If the operators A and B have the same signature under time reversal, and the Hamiltonian is invariant under time reversal, it is easy to see that (A˙|B) = 0. Hence (P˙i|Pj) = 0, which simplifies the memory matrix. The dominant contributions to σ(ω) come from the slowly relaxing modes, which are Px,y. Using the invariance of the Hamiltonian under (x, y) → (−x, y), the expression for the DC diagonal conductivity reduces to, to leading order in the perturbing Hamlitonian [97, 144]:

( )− i 1 σ = lim |(J |P )|2 P˙ P˙ , (2.15) xx → x x x − x ω 0 ω L0 0

where the subscript 0 denotes evaluation with respect to the unperturbed Hamiltonian. We then have

∫ β χJP = (Jx|Px) = dτ⟨Jx(τ)Px(0)⟩, 0 2 |χJP | σxx = lim ∫ ∞ , ω→0 1 iωt ˙ ˙ ω dte [Px(t), Px(0)] [ ] 0 Im[GR (ω)] 1 P˙xP˙x ρxx = Re = lim 2 . (2.16) σ ω→0 ω|χJP |

We compute the χJP susceptibility for L in Appendix B.1. There we find that although the continuum limit hot-spot theory with linearized fermion dispersion has χJP = 0, upon including Fermi surface curvature we have χJP ≠ 0, even at T = 0. We will henceforth assume that χJP is a T -independent non-zero constant.

∼ −2 However, if χJP is small, then the DC resistivity ρ(T ) χJP will be large, and there will be a crossover to a higher T regime where we have to consider the physics of a system with χJP = 0: note that it is possible for such a system to have a non-zero resistivity even in the absence perturbations which relax momentum.

Important, previously studied examples of theories with χJP = 0 are conformal field theories [102, 145–

147] and it would be interesting to extend such studies to the quantum-critical spin density wave theory

[86, 89].

62 We also see that χSP = 0 for the spin current due to the spin rotation symmetry of the model.

2.4 Contributions to the DC Resistivity

In this section, we compute the contributions to the DC resistivity ρxx(T ) coming from the random-mass term and from the scattering of hot spot fermions by the random potential. To apply the memory matrix formalism, we compute the time dependence of the conserved momentum arising from the perturbations in

LV + Lm. Using P˙x = i[H,Px], we obtain

∫ [ ] d2q d2k P˙ = −i k V (k)ψ†(q + k)ψ(q) + V (k)ψ†(q + k)Γ ψ(q) + m(k)ϕ (q)ϕ (−q − k) , x (2π)4 x 1 2 0 µ µ (2.17) giving [∫ ] 2 ( ) R d k 2 2 R R 2 R Im[G ˙ ˙ (ω)] = Im kx V0 (Ξ1 (k, ω) + Ξ2 (k, ω)) + m0Π (k, ω) , (2.18) Px,Px (2π)2

R † † R where Ξ1,2 are the retarded Green’s functions for ψ ψ and ψ Γ0ψ respectively and Π is the retarded Green’s function for ϕµϕµ.

2.4.1 Random-Mass Term

We use the following form for the vector boson propagator, which is derived in Appendix B.2:

δ D (q, iω ) = µν , (2.19) µν q 2 2 | | q + ϵωq + γ ωq + R(T ) where R(T ) is a positive-definite mass term at finite temperature which is computed in Appendix B.2. The

Green’s function for ϕµϕµ may be obtained by resumming the graphs shown in Fig. 2.2; these are precisely the graphs that have to be summed at leading order in a large N expansion in which ϕµ has N components.

63 We obtain 2Π(˜ k, iΩ) Π(k, iΩ) = , (2.20) 1 − (20/3)uΠ(˜ k, iΩ) where ∫ ∑ d2q Π(˜ k, iΩ) = T D (q, iω )D (q + k, iω + iΩ). (2.21) (2π)2 µν q νµ q ωq

Figure 2.2: Resummation of graphs to obtain the Green’s function for ϕµϕµ. The diamonds denote ϕµϕµ operators and the circles denote the quartic interaction. The wavy lines represent the vector boson propagators.

Then we have, for large u,

1 9 Im[Π˜ R(k, ω)] lim Im[ΠR(k, ω)] = lim , (2.22) ω→0 ω ω→0 200u2ω Re[Π˜ R(k, ω)]2

The z = 2 regime may be accessed by sending ϵ → 0 with γ ≠ 0. Then we have (See Appendix B.3 for computations)

∫ [ ( )] 2 Λ 2 2 3 2 m0 d k 2 R ≈ m0γ T Λ ρxx(T ) = lim 2 2 kxIm[Π (k, ω)] 2 2 c1 + c2 ln , (2.23) ω→0 ω|χJP | (2π) u |χJP | γT

where Λ is a momentum cutoff that is much larger than any other scale in the problem, and c1, c2 have only very slow log-log dependences on T .

In the γ → 0 limit with ϵ ≠ 0, z = 1. In this regime, all the momentum integrals involved converge (See

64 Appendix B.3). We get

( ) Im[Π˜ R(k, ω)] 1 k2 lim = F , → 2 2 ω 0 ω ϵT ( ϵT) 1 k2 lim Re[Π˜ R(k, ω)] = G . (2.24) ω→0 ϵT ϵT 2

′ Thus we can cast the integral for ρxx(T ) in terms of a dimensionless momentum k and obtain

∫ 2 3 4 2 ′ ′2 2 3 4 9m0ϵ T d k ′2 F (k ) m0ϵ T ρxx(T ) = 2 2 2 kx ′2 2 = 2.42 2 2 . (2.25) 200u |χJP | (2π) G(k ) u |χJP |

We also obtain a temperature driven crossover in the scaling of ρxx(T ) when both ϵ ≠ 0 and γ ≠ 0. We have ρ(T ) ∼ T in the z = 2 regime at low T and ρ(T ) ∼ T 4 in the z = 1 regime at high T , as shown in

Fig. 2.3. The T 4 behavior agrees with Eqs. (2.6,2.7) with the large N value of the exponent ν = 1.

4 Figure 2.3: Temperature driven crossover in the scaling of the random-mass contribution to ρxx(T ) from T to T as T is increased. Here, γ = 1, ϵ = 1 and the momentum cutoff Λ = 100.

2.4.2 Fermionic Contributions

Since the boson couples most strongly to the fermions near the hot spots, we expect the most significant non

Fermi liquid contributions to the resistivity to come from the scattering of these hot spot fermions by the

65 random potential and not involve the cold fermions elsewhere on the Fermi surfaces. The random potential can scatter these fermions between hot spots, which results in a large momentum transfer, or within the same hot spot, with a much smaller momentum transfer. Since the expression for the resistivity contribution contains a factor of k2, we expect the contributions due to inter hot spot scattering to be much larger than those due to intra hot spot scattering.

Considering pairs of hot spots (i, j), i ≠ j separated by vectors Qij in momentum space, we expand the momentum k transferred by the random potential about Qij in Eq. 2.18 to obtain, to leading order, the contribution to G from inter hot spot scattering P˙x,P˙x

∫ [ ∑ 1/T ⟨ ⟩ † † G (iΩ) = V 2 Qij2 dτ ψ (r = 0, τ)ψ (r = 0, τ)ψ (r = 0, 0)ψ (r = 0, 0) + P˙x,P˙x 0 x j i i j ̸ 0 i,j,i=j ] ⟨ ⟩ † 0 † 0 iΩτ ψj (r = 0, τ)Γ ψi(r = 0, τ)ψi (r = 0, 0)Γ ψj(r = 0, 0) e , (2.26)

( ) where the subscripts now denote that the fermions belong to a particular hot spot, i.e. ψ = ψia and ψ i ψib ia,ib are two-component spinors. This leads to the graphs shown in Fig. 2.4. The fermion dispersions are now linearized about the hot spots:

ξiα(k) = viα · k. (2.27)

The first (free fermion) graph in Fig. 2.4(a) gives

∫ 2 2 ∑ ∑ V 2Λ2 ∑ ∑ ij2 − V0 Λ ij2 dξiα ′ 0 ∥ Qx ρxx(T ) = 2π 2 Qx 4 nF (ξiα) = 3 2 , (2.28) |χJP | (2π) viαviβ (2π) |χJP | viαviβ i,j,i≠ j α,β i,j,i≠ j α,β which is simply a temperature-independent constant. Here the indices α, β run over the two fermion types

ij a, b, and Λ∥ ≪ Q is a cutoff for the momentum components parallel to the Fermi surfaces at the hot spots. ∫ The subsequent graphs in Fig. 2.4(a) all contain factors of the form dξ/(iω − ξ)m, where m is an integer

66 Figure 2.4: Graphs for the contribution to G (iΩ) due to inter hot spot scattering. The vertices provide factors P˙x,P˙x ij ij of Q V0 + Q V0Γ0. The solid lines are fermion propagators and the wavy lines are vector boson propagators. The dotted lines carry internal momentum and the external bosonic Matsubara frequency iΩ, and have propagators equal to 1. The first graph in the series of graphs in (a) is the free fermion contribution. The subsequent graphs represent the∫ corrections due to renormalization of the fermion propagators at one loop, but evaluate to 0 due to factors of dξ/(iω − ξ)m = 0, m ∈ Z and m ≥ 2. The graph in (b) is the simplest vertex correction. Here too, for the same reason, further graphs of the same type but with self-energy rainbows on the fermion propagators also evaluate to 0.

≥ 2, coming from the fermion propagators separated by self energy rainbows, and hence evaluate to zero.

The leading vertex correction is given by the graph in Fig. 2.4(b). Again, for the same reason, we can get

away with using the bare fermion propagators instead of the one loop renormalized ones. We compute this

correction in Appendix B.4. In the z = 2 limit, we obtain

T ρ (T ) ∼ const. + bT + c , (2.29) xx ln(Λ2/(γT )) which also contains terms that scale linearly in T . In the z = 1 limit, we have

′ ρxx(T ) ∼ const. + b T, (2.30)

which is still linear in T . Other corrections whose graphs contain fermion loops connected by boson propa-

gators are less significant: due to momentum conservation at each vertex, some of these boson propagators

must carry a large momentum of the order of Qij, hence suppressing their contribution. Also, graphs having

67 a single fermion loop that runs through both the external vertices, but containing multiple boson propagators which could be attached in any way, will always have the aforementioned factors that evaluate to zero once all the boson momenta and frequencies are set to zero, thus suppressing their most singular contributions.

2.5 Discussion

We have proposed a perspective on DC transport in the vicinity of a spin-density wave quantum critical point in two dimensional metals; the results can also apply to other density wave transitions of metals in two dimensions. Whereas previous perturbative approaches[90, 112] started from a quasiparticle picture which eventually breaks down at hot spots on the Fermi surface, we have argued for a strong-coupling perspective in which no direct reference is made to quasiparticles. Instead, we assume that strong interactions cause rapid relaxation to local thermal equilibrium, and the flow of electrical current is determined mainly by the relaxation rate of a momentum which is conserved by the strong interactions. We used weak disorder as the primary perturbation responsible for momentum relaxation, and then obtained a formally exact expression for the resistivity in terms of two-point correlators of the strongly-interacting and momentum-conserving theory.

Our final results were obtained by an evaluation of such two-point correlators. Here, we used a simple large N expansion, and found a resistivity that varied linearly with T . Clearly, an important subject for future research is to evaluate these correlators by other methods which are possibly more reliable in the strong-coupling limit.

Our computations also found distinct sources for the residual resistivity and the co-efficient of the linear

T term in the resistivity. The residual resistivity is entirely fermionic, and arises from scattering between well-separated points on the Fermi surface, induced by short-wavelength disorder. In contrast, the linear resistivity has a bosonic contribution from long-wavelength disorder. Moreover, the latter can be strongly

68 enhanced in systems with small χJP , the cross-correlator between the total momentum and the total current.

For experimental applications, BaFe2(As1−xPx)2 offers probably the best testing ground so far for our theory: this material has a spin density wave quantum critical point near x = 0.3, and a clear regime of linear-in-T resistivity above it [7, 9]. It would be interesting to carry out these experiments while carefully reducing the degree of long-wavelength disorder, including grain boundaries and dislocations. Our theory implies that the co-efficient of the linear-in-T resistivity should decrease in such sample. Note also our argument above that the residual resistivity cannot be used as a measure for the degree of disorder (as is often done); the residual resistivity is mainly sensitive to short-wavelength disorder.

69 It is life, I think, to watch the water. A man can learn so many things.

Nicholas Sparks

3 Hydrodynamic flows of non-Fermi liquids:

magnetotransport and bilayer drag

3.1 Introduction

Hydrodynamic flow of electrons can occur in solid state systems provided that the microscopic length scale of momentum-conserving electron-electron collisions is sufficiently short [32]. Under this condition the electron liquid attains local equilibrium and can be described in terms of slow variables associated with conserved quantities such as momentum and energy. However, this transport regime was hard to realize experimentally as typically electron-impurity scattering degrades electron momentum, whereas electron-phonon collisions

70 violate both momentum and energy conservations of the electron liquid. Early evidence for the so-called hydrodynamic Gurzhi effect, related to the negative temperature derivative of resistivity, was reported in thin potassium wires [148], and later in the electrostatically defined wires in the two dimensional electron gas of (Al,Ga)As heterostructures [149]. The recent surge of experiments devoted to revealing hydrodynamic regimes of electronic transport is mainly focused on measurements conducted on graphene [34, 150].

In the context of transport theories, a hydrodynamic description is powerful as it accurately describes most liquids. All microscopic details of the system at hand are then encoded into a handful of kinetic coefficients such as viscosities and thermal conductivity. In certain cases the latter can be controllably derived from the linearized Boltzmann kinetic equation by following the perturbative Chapman-Enskog procedure developed originally for gases. However, we have examples now where this kind of microscopic approach has to be substantially revisited. Deriving hydrodynamics for linearly dispersing electronic excitations in graphene represents an interesting example where this standard computation scheme had to be redone from scratch

[151–155]. An even more dramatic example is given by strongly correlated electron liquids [156], where the effects of interactions are nonperturbative, and thus a Boltzmann-like description may not be applicable.

Yet the hydrodynamic picture still holds [157] and has to be viewed as a phenomenology that enables one to express various transport observables in terms of pristine kinetic coefficients of the electron liquid and certain thermodynamic quantities. This is our motivation to consider a hydrodynamic description of transport for strongly correlated electron liquids where we do not assume Fermi liquid-like behavior. We also do not assume Galilean invariance to be present. In this study we focus on magnetotransport and frictional drag transresistance in bilayers.

71 3.2 Hydrodynamic formalism

The general linearized set of equations that govern nonrelativistic magnetohydrodynamic transport in two dimensional charged fluids are given by [60, 69, 95, 158, 159] (i) the force equations (repeated indices imply summation throughout this chapter)

∂t(Mvi) + ∂jTij = QEi + Sξi + BϵijJj,

Tij = P δij − η(∂ivj + ∂jvi) − (ζ − η)∂kvkδij, (3.1)

which relate the rate of change of the momentum density to pressure, viscous, thermoelectric and Lorentz forces. M serves as an effective “mass density” and Q is the effective charge density of the fluid. η and ζ respectively are the shear and bulk viscosities. Ei and ξi represent the electric field and thermal gradient.

Fluctuations in the fluid pressure P are given by dP = Qdµ + SdT , where S is the entropy density and µ is the local screened chemical potential per unit charge. (ii) The equations for charge and heat currents read respectively as

− Q − − − Q − Ji = Qvi σij (∂jµ Ej Bϵjkvk) αij(∂jT ξj),

H − Q − − − Q − Ji = T Svi T α¯ij(∂jµ Ej Bϵjkvk) κ¯ij(∂jT ξj), (3.2)

where σQ, αQ and κ¯Q are microscopic “incoherent” conductivities [160], and (iii) the continuity equations are

H ∂tQ + ∂iJi = ∂t(TS) + ∂iJi = 0. (3.3)

Q Q − Onsager reciprocity requires αij(B) =α ¯ji( B). The incoherent conductivities, viscosities and thermodynamic properties are derived from correlation functions of the underlying microscopic field theory of the

72 non-Fermi liquid [161–163]. This is a generalization of the usual theory of hydrodynamics to systems without Galilean invariance.

3.3 Magnetotransport in a single layer

We consider the steady state solutions of these equations in the presence of a disordered chemical potential

µ(x). In the absence of applied electric fields and temperature gradients, we can apply a background electric field E¯i = ∂iµ to nullify currents and fluid motion, assuming a uniform temperature. We then look for steady state solutions when this background is perturbed by an infinitesimal uniform electric field δEi in linear response [95, 155]. The difference between the unperturbed and perturbed set of equations gives

− Q − − − Q ∂iJi = ∂i(Qvi σij (∂jδµ δEj Bϵjkvk) αij∂jδT )

− Q − − − Q =∂i(T Svi T α¯ij(∂jδµ δEj Bϵjkvk) κ¯ij∂jδT )=0,

Q(∂iδµ − δEi) + S∂iδT − ∂j(η(∂ivj + ∂jvi)) − ∂i((ζ − η)∂kvk) = BϵijJj, (3.4)

where the delta-quantities represent deviations from the background values generated by the applied electric field (vi ∼ O(δ)). We read off transport coefficients by looking at the change of the uniform components of their respective currents with respect to the applied electric field. For example, σxx = δJx(q = 0)/δEx and σyx = δJy(q = 0)/δEx. The equations (3.4) need to be given periodic boundary conditions in order to ensure a unique solution; otherwise one may shift v by a constant and cancel the effects by appropriately shifting δµ by a function that has a constant gradient [95]. We can consider the solution of (3.4) while treating disorder perturbatively [95, 155]. Against a uniform background chemical potential, it is easy to see

(0) (0) (0) that the only response is a finite uniform velocity field vi = ϵijδEj/B, which implies σij = ϵijQ /B, where Q(0) is the uniform charge density in the absence of any disorder. Introducing a small parameter ϵ

73 ∑ ∞ n (n) to parameterize the strength of the disorder, we expand µ(x) = n=1 ϵ µ (x). All responses, densities, viscosities and microscopic conductivities may also be expanded in powers of ϵ. For example vi(x) = ∑ ∑ ∞ n (n) ∞ n (n) n=0 ϵ vi (x) and Q(x) = n=0 ϵ Q (x).

(n) (n) (n) Order by order in ϵ, there are 4 unknowns δµ , δT , vi and 4 equations in (3.4), so a unique solution is possible. This expansion in disorder strength while keeping B finite implies the assumption that the magnetic field relaxes momentum faster than the disordered potential (see [164] for when both relaxation rates are comparable). The expression for the uniform charge current at O(ϵ2) is (in momentum space)

∫ (2) (2) − (1) − (1) (1) − (1) BJi (k = 0) = ϵijQ (k = 0)Ej iϵij (Q ( k)δµ (k) + S ( k)δT (k))kj. (3.5) k

Thus, solving the equations at O(ϵ) gives all the information needed to obtain the uniform conductivities up to O(ϵ2).

In general the densities, viscosities and incoherent conductivities depend on B, and their functional forms can be deduced from the underlying quantum critical theory, which is beyond the scope of hydrodynamics.

However, for small values of B these dependences can be neglected as the dominant effect on magnetoresistance arises from the long-range modulations of the equilibrium density (see Refs. [165–167] for other large B effects). This contribution exceeds the one due to the B-dependence of the kinetic coefficients of the liquid by a parametrically large factor controlled by the ratio of disorder wavelength to electron equilibration length. We hence set the off-diagonal components of the quantum critical transport to zero. We assume that ∂Q/∂µ ≠ 0 and ∂S/∂µ ≠ 0, so Q(1) ≠ 0 and S(1) ≠ 0. The solutions of (3.4) are provided in the

Appendix C.

2 Using (3.5) to read off the uniform charge current, we see that σxx,yy are O(ϵ ), whereas σxy is O(1).

≈ 2 Hence the symmetrized electrical resistance is given by Tr ρ (σxx + σyy)/σxy. This is in general a very complicated function, with a potentially complicated temperature dependence due to the temperature depen-

74 dences of all the microscopic coefficients. However, if we assume that the disorder is very long wavelength, thus retaining only the leading contribution in the inverse disorder wavelength in the diagonal conductivity, we find a rather simple result

∫ Q(0) + ϵ2Q(2)(k = 0) ϵ2 k k |Q(1)(k)|2 σ = σ(0) + ϵ2σ(2) = ϵ + ϵ ϵ l m , ij ij ij ij (0) il jm 4 (3.6) B η k k

which is consistent with Onsager reciprocity σij(B) = σji(−B). All corrections from the microscopic incoherent conductivities appear at higher orders in the inverse disorder wavelength (for details see Appendix C).

For the second term of (3.6) to be smaller than the first, so the perturbative structure is consistent, we must

≪ (0) (0) 4 1/2 have (∂Q/∂ ln µ) (η Q )/(λµB)) , where λµ is a characteristic wavelength of the disorder. To leading order in ϵ, one gets the symmetrized magnetoresistance at leading order in the inverse disorder wavelength

∫ ϵ2B2(∂Q/∂µ)2 |µ(1)(k)|2 ρ(B) − ρ(0) = T . Tr Tr (0)2 (0) 2 (3.7) Q η k k

The temperature dependence of the magnetoresistance is controlled only by the viscosity in this long-wavelength disorder limit, as was the case in [168] for the special case of Galilean-invariant fluids (σQ = αQ = 0).

However, there the magnetoresistance was controlled only by the viscosity regardless of the spectrum of the disorder. Since we do not expect most non-Fermi liquid metals to be Galilean-invariant, this is an important strengthening of the previous result. It can additionally be shown that the long-wavelength disorder result

(3.7) is also insensitive to the Hall viscosity [169] and vorticity susceptibility [170], which are new parity-odd microscopic transport coefficients that can appear in the presence of a magnetic field.

The above result could enable the extraction of the temperature dependence of the viscosity of the electron liquid as δρ(B)/ρ(0) ∝ 1/η(0) and thus allow for testing theoretical models of potential non-Fermi liquid states in the hydrodynamic regime. In Appendix C we also provide results for the magnetothermal resistance.

75 3.4 Drag transport in bilayers

For drag type transport [171], we use our hydrodynamic equations for each layer of the bilayer system, with E = B = ξ = 0. Drag is generated by intrinsic hydrodynamic fluctuations encoded in fluctuating

H 1,2 → 1,2 noise terms [33, 172–174] added to Tij,Ji,Ji that are uncorrelated between the layers (Tij Tij +

1,2 1,2 → 1,2 1,2 H 1,2 → H 1,2 1,2 sij , Ji Ji + ri , Ji Ji + gi )

⟨ 1,2 1,2 ′ ′ ⟩ (0) sij (k, ω)slm (k , ω ) = 2T (η (δilδjm + δimδjl)

(0) (0) ′ ′ + (ζ − η )δijδlm)δ(k + k )δ(ω + ω ), (3.8)

⟨ 1,2 1,2 ′ ′ ⟩ Q(0) ′ ′ ri (k, ω)rj (k , ω ) = 2T σ δijδ(k + k )δ(ω + ω ),

⟨ 1,2 1,2 ′ ′ ⟩ 2 Q(0) ′ ′ gi (k, ω)gj (k , ω ) = 2T κ¯ δijδ(k + k )δ(ω + ω ),

⟨ 1,2 1,2 ′ ′ ⟩ 2 Q(0) ′ ′ ri (k, ω)gj (k , ω ) = 2T α δijδ(k + k )δ(ω + ω ),

with all other correlators of the sources being zero. These fluctuations induce fluctuations in the charge and

(0) (0) → (0) (0) entropy densities in the layers (Q1,2,S1,2 Q1,2 + δQ1,2,S1,2 + δS1,2). The fluctuations of chemical potential and temperature are expressed in terms of the charge and entropy fluctuations

( ) ( ) ∂µ ∂µ δµ1,2 = δQ1,2 + δS1,2, (3.9) ∂Q S ∂S Q

and likewise for δT1,2. We must add to the pressure term in each layer the effects of intra and inter-layer

Coulomb forces generated by the fluctuations in the charge densities (the layers are separated by a distance d) 2πQ(0) δP → δP + (δQ + e−kdδQ ). (3.10) 1,2 1,2 k 1,2 2,1

76 The drag resistance measures the sensitivity of the electric field induced by the dragging force in the open- circuit passive layer to the current flowing in the driven layer. It is given by

E F (v ) − F (0) ρ ≡ 2D = 12 x 12 , D (0)2 (3.11) J1 Q vx ∫ 2π −kd F12(vx) = −i kxe ⟨δQ1(k, ω)δQ2(−k, −ω)⟩. k,ω k

The derivation of these force and pressure relations only requires a straightforward application of Coulomb’s law. In additional to the noise sources, we also linearize in the velocity vx (the driven layer is driven by this

(0) uniform velocity field, not by an electric field). Note that J1 = Q vx is valid even for non-Galilean invariant fluids as the noise terms themselves cannot induce any uniform current flow due to averaged inversion

(0) and time-reversal symmetries. Thus J1 must vanish when vx = 0, and renormalizations of Q due to the noise terms are subleading.

We neglect the effects of thermal currents: they produce only subleading effects at large spatial separations

1 2 (see Appendix C for further details). Switching to the basis defined by δQ = δQ1 δQ2, sij = sij sij,

1 2 ri = ri ri , the hydrodynamic equations can be reduced to the form

M 2 ΠδQ + k (i(D + D )k + ω)v (δQ + δQ−) = 2 x σ η x + M k v k (r+ + r−) − iM(D k2 − iω)k r − Q(0)k sk , 2 x x i i i η i i i ij j

2 2 (0) (0) −kd Π = M(Dηk − iω)(Dσk − iω) + kQ (2πQ (1 e ) + b1k), (3.12)

Q(0) (0) (0) (0) where Dσ = σ (∂µ/∂Q)S, Dη = (η + ζ )/M and b1 = Q (∂µ/∂Q)S. The solutions to these

(0) (1) equations are linearized in vx: δQ = δQ + δQ vx. Since the vx-less configuration obeys averaged

(0) inversion and time-reversal symmetry and vx always appears as vxkx which is odd under inversion, δQ

77 (1) is even under k → −k whereas δQ is odd. The dragging force may be written as

∫ − F12(vx) F12(0) kxvx ⟨ (0) (1) − − ⟩ − ⟨ (0) (1) − − ⟩ = kd ( δQ+ (k, ω)δQ− ( k, ω) δQ− (k, ω)δQ+ ( k, ω) ). (3.13) iπ k,ω ke

All other terms vanish upon momentum/frequency integration due to even/odd cancellations. Inserting the

σ η σ η solutions of (3.12), we obtain ρD = ρD + ρD, where ρD is generated by the charge fluctuations r and ρD is generated by the viscous fluctuations s:

∫ ∫ ∞ 7 2 − 2 4 σ 3 Q(0) (Dσ + Dη)k (ω Dηk ) ρD = M T σ dk 2kd 2 2 , e |Π+| |Π−| 0 ω ∫ ∫ ∞ 9 η (0)2 (0) (0) (Dσ + Dη)k ρD = MQ T (η + ζ ) dk 2kd 2 2 . (3.14) 0 ω e |Π+| |Π−|

This yields a complicated integral expression for ρD. We can however make simplifications in the regimes of “large” and “small” d. The model of Fermi surface coupled to U(1) gauge field has roughly the following properties [31, 161, 162] for dynamical critical exponent z = 3 (m is the effective fermion mass), corresponding to the case of short-ranged interactions of composite fermions [31]:

( ) ∂µ ℏ2 E Q(0) ∝ ek2 ,M ∝ mk2 , ∝ , b ∝ F , F F ∂Q e2m 1 e ( ) S ( ) 2/3 2 2/3 Q(0) ∝ EF e (0) ∼ (0) ∝ EF ℏ 2 σ , η ζ kF . (3.15) kBT ℏ kBT

3 ≫ 3 ≡ 2 (0)2 d is said to be “large” when d dc M(Dη + Dσ) /Q . This gives

( ) 4/3 3 ≫ EF εEF (kF d) 2 . (3.16) kBT e kF

We have set the electrostatic permittivity ε = 1 so far in the work but restored it in the last equation. We

2 2 also demand d ≫ de ≡ ℏ ε/(e m), which is trivially achieved as de is typically a very small distance scale

78 0.36 0.012

0.27 0.009

0.18 0.006

0.09 0.003

0 0 0 1 2 3 4 5

σ η Figure 3.1: Normalized drag resistance ρD(T )/ρ0(d). ρD(T ) = ρD(T ) + ρD(T ) is obtained by numerically evaluat- 2 4 ing (3.14) for two different spatial separations. ρ0(d) = (ℏ/e )/(kF d) . Note the crossover from positive to negative curvature as d is increased. This feature holds for other values of the dynamical critical exponent 2 < z < 3 as well that can appear in the theory of Halperin, Lee and Read. We use TF ∼ 50 K and m ∼ me/4 (dc ∼ 10 nm at T ∼ 5 K). We set all constants of proportionality in (3.15) to 1. Numerical values should be treated as order-of-magnitude estimates only.

−10 (O(10 ) m for m ∼ me/4).

For d ≫ dc we obtain the leading contributions

( ( ) ) dk2/3 4/9 ( ) 4 F kB T 1/3 ln ℏ 1/3 EF σ ∼ kBT de ρD 2 4 , e EF (kF d) ( ( ) ) dk2/3 4/9 ( ) 5 F kB T 1/3 ln ℏ 1/3 EF η ∼ kBT de ρD 2 5 . (3.17) e EF (kF d) /(kF de)

σ η η ρD and ρD have the same temperature scaling up to logarithms. However, ρD falls off faster with d than

σ ρD. This results should be contrasted to that obtained earlier for Fermi liquids [173]. Note that even though the power dependence on temperature is T 1/3, there is a ln4(T ) correction, which will make the temperature dependence appear faster than T 1/3 but slower than T , which is consistent with the data of Refs. [175, 176] at large separations.

n>1 At small separations d ≪ dc, all contributions to ρD scale as T (see further details in C). This is again consistent with [176], which shows an apparent crossover from positive to negative curvature in ρD(T ) as a function of T as d is increased. In Fig. 3.1 we show ρD(T ) obtained by numerically evaluating the integrals

79 without the above approximations that confirm the qualitative behaviors we discussed. It should be carefully noted that in Fig. 3.1 the line corresponding to d = 150 nm appears superficially above the line of d = 15 nm plot which is due to the choice of the normalization factor ρ0(d). Drag is obviously a decaying function of inter-layer separation d as is clear from (3.17).

3.5 Discussion

The most extensively studied example of transresistance in the case of non-Fermi liquids corresponds to inter-layer frictional Coulomb drag between bilayers of half-filled Landau levels [175–178]. The theoretical approach that has proved most useful for understanding the filling fraction ν = 1/2 state is the fermion Chern-

Simons field theory, which is based in turn on the composite-fermion picture [31]. Previous calculations

4/3 [179–181] showed that the dominant low-temperature behavior for ρD scales with temperature as T (see

Appendix C for a brief summary of this result). This unique power exponent can be traced back to a special momentum dependence of the electronic longitudinal conductivity, as can be deduced from surface acoustic wave measurements. Indeed, in the composite-fermion picture, at ν = 1/2, the density response at small

3 −1 frequencies and small wave-vectors is of the form ∝ (k − 8πiχωkF ) , which can be viewed as slow diffusion with an effective diffusion constant that vanishes linearly with k (where χ is the thermodynamic compressibility of the ν = 1/2 state). Since the typical frequency is set by temperature ω ∼ T , the pole structure of long-wavelength density fluctuations sets a characteristic scale for momentum transfer between

1/3 4/3 the layers k ∝ T that then carriers over to drag resistance ρD ∝ T . This should be contrasted the

2 1/3 Fermi liquid prediction ρD ∝ T at lowest temperatures, and our prediction ρD ∝ T . In our current understanding, the results of [179] correspond to the “collisionless” regime of transport with respect to intralayer collisions, namely a long equilibration length as mediated by interactions with the gauge field. We considered the opposite collision-dominated regime where this length scale is assumed to be short. This

80 should explain the difference between the power exponents 4/3 and 1/3 between two limiting cases. We hope that understanding different transport regimes and corresponding temperature dependencies will be of help for the interpretation of future experiments, as it also deepens our current understanding of the existing transport data and corresponding theories.

81 No one means all he says, and yet very few say all they mean, for words are slippery and thought is viscous.

Henry Brooks Adams

4 Shear viscosity at the Ising-nematic quantum

critical point in two dimensional metals

4.1 Introduction

Recent experiments on graphene [34, 150] and PdCoO2 [182] have displayed remarkable evidence for nearly- momentum-conserving hydrodynamic flow of the electron liquid. In clean Fermi liquids, hydrodynamic flow requires very clean samples with weak umklapp scattering so that electron-electron collisions lead to thermalization before there is significant momentum lost to the crystal [32, 149, 157, 182]. However, rapid thermalization and hydrodynamics are natural properties of quantum critical systems [102] and strange

82 metals [60, 62, 66, 69], and their consequences should be visible even in moderately clean samples. Graphene was proposed as a strange metal in which ill-defined quasiparticles lead to hydrodynamic flow at intermediate temperatures [108, 151, 152, 155, 158, 183–189]: the experiments also display evidence [34, 150, 155, 187,

188] for the viscous drag of such flow. There have also been studies of viscous flow in high energy physics

[190–193] and ultracold atoms [194–198].

These experimental advances indicate that the time is ripe for exploring hydrodynamic electron flow in the ubiquitous strange metal regimes of the cuprates or the pnictides. These are metals without quasiparticle excitations, and so should exhibit hydrodynamic flow when impurities are dilute. We note the indirect evidence for such behavior in the photoemission experiments of Rameau et al. [199]. To this end, here we examine the simplest model which realizes a metallic state without quasiparticles in two spatial dimensions, and compute its shear viscosity, η. We will study the quantum critical point (QCP) for the onset of Ising nematic order [74, 200, 201] using its continuum field theoretic formulation using patches on the Fermi surface

[28, 79].

General scaling arguments (reviewed below) for a spatially isotropic system imply that η should scale in the same manner as the entropy density, s; so

η/s ∼ ℏ/kB, (4.1)

where the r.h.s. restores dimensions, and the prefactor is expected to be of order unity. (In d = 2 hydrodynamic long time tails can lead to logarithmic corrections to η [33] which we ignore here, as we find much larger corrections). This is also the expectation from holographic studies of critical quantum liquids

[190, 191, 202–206]. A relationship of the form (4.1) appeared in string-theoretic realizations of strongly- coupled field theories [191], and has been widely used as a diagnostic of strongly-coupled non-quasiparticle dynamics in the quark-gluon plasma [190–193].

83 Figure 4.1: Fermi surface and definition of the momentum components parallel (k∥) and perpendicular (k⊥) to the Fermi surface at the two Fermi surface patches in which the low-energy field theory is defined.

Our main result is that Eq. (4.1) does not apply to many of the models of electronic strange metals without quasiparticles. Even without long-lived quasiparticles, such models have a Fermi surface at T = 0, which defines momenta with singular low energy excitations; more precisely, the Fermi surface is the locus of points at which the inverse Green’s function vanishes. Although the metal is globally isotropic, the excitations in the vicinity of a particular point on the Fermi surface have a highly anisotropic structure, as shown in

z Fig. 4.1: excitations at a momentum k⊥ perpendicular to the Fermi surface have a typical energy k⊥, while

2z excitations at a momentum k∥ parallel to the Fermi surface have a typical energy k∥ ; here z is the dynamic critical exponent. In the present chapter, we will show that the dispersion of the excitations parallel to the

Fermi surface plays a more fundamental role in determining the value of the shear viscosity η. As a result

Eq. (4.1) does not apply, and we find instead a divergence as T → 0,

η/s ∼ T −2/z. (4.2)

This surprising violation of (4.1) in an isotropic system can be traced directly to the presence of a Fermi surface. Our result implies that holographic duals of strange metals [83, 207–211] do not fully capture the

Fermi surface structure. Instead, it appears that bulk quantum gravity corrections will be required to resurrect the Fermi surface in the holographic theories [212–214], and to obtain the result corresponding to Eq. (4.2).

84 Section 4.2 will present a review of scaling arguments which usually apply the conventional relation in Eq. (4.1). The dimensionally extended field theory for the quantum critical point will be presented in

Section 4.3. We will use this field theory to compute the ‘optical’ shear viscosity (i.e. the viscosity at frequencies ω ≫ T ) in Section 4.4. We will then examine the usual DC viscosity (at frequencies ω ≪ T ) in

Section 4.5.

4.2 Scaling arguments

In studies so far of the thermodynamic and transport properties of strange metals, the anisotropy of the

Fermi surface has had a specific consequence [161]: the entropy density, and the electrical and thermal conductivities are dominated by the energy dispersion perpendicular to the Fermi surface, while the direction parallel to the Fermi surface mostly acts as a label which counts the total density of perpendicular excitations.

Consequently, in scaling arguments we find a violation of hyperscaling: this is the property in which the entropy density of a d dimensional system scales as if it is in d − θ dimensions, with θ the violation of hyperscaling exponent. For a Fermi surface θ = d − 1, because only the dispersion perpendicular to each point on the Fermi surface is important in the computation of the entropy. Recent work has shown [161] that similar arguments also correctly determine the electrical conductivity and entropy density.

We now review the general scaling arguments for the universality of η/s. The entropy density invariably scales as a density, and so has scaling dimension d. From the arguments just presented above, with the violation of hyperscaling in the presence of a Fermi surface, the entropy density s should have scaling dimension d − θ, and so

s ∼ T (d−θ)/z; (4.3)

this was confirmed by computations in [161]. Similar arguments apply to the optical conductivity σQ(ω), where ω is a frequency; naively, the conductivity has scaling dimension d − 2, and so we can expect that in

85 the presence of a Fermi surface, the dimension will be d − θ − 2. The computation in [161] shows that this is indeed the case, and we have

(d−2−θ)/z σQ ∼ T Υ(ω/T ), (4.4) where Υ is a scaling function.

In an isotropic system that obeys hyperscaling, we can read off the scaling dimension of the stress tensor from its definition [215],

( ) ∑ δL δL T = ∂ ζ − ∂ ∂ ζ − δ L, (4.5) µν δ(∂ ζ ) ν n µ δ(∂2ζ ) ν n µν n µ n µ n

where ζn denotes all the fields in the theory and L the Lagrangian density. It follows that the spatial components have the same scaling dimension as the Lagrangian density, [Tij] = d + z, and that the mixed temporal-spatial components have scaling dimension [T0i] = d + 1. Inserting these scaling dimensions into the Euler equation,

∂tpα = ∂βTαβ, (4.6)

where pα are the components of the momentum operator, yields consistent results. We thus obtain

[Txy] = d + z (4.7)

in the presence of hyperscaling. Kubo’s formula for the frequency-dependent shear viscosity reads [195, 196],

−1 Re η(ω) = lim ω Im χT T (ω, q), (4.8) q→0 xy xy where

⟨ − − ⟩ χTxyTxy (iωn, q) = Txy(iωn, q)Txy( iωn, q) (4.9)

86 is the autocorrelation function of the xy-component of the stress tensor T . Its scaling dimension is

[η] = −z − d − z + 2[Txy] = d (4.10)

and the d.c. shear viscosity is given by η = limω→0 η(ω). This is the same scaling dimension as for the entropy density above. With the violation of hyperscaling in the presence of a Fermi surface, the examples of the entropy density and the optical conductivity above suggest that η should scale just like s in Eq. (4.3), and hence Eq. (4.1) should apply. Our computations in this chapter show that this is not true, and the Fermi surface leads to behavior genuinely different both from naive scaling assumptions, and from holographic examples: the T dependence of η is such that Eq. (4.2) holds.

4.3 Field theory

We now recall the field theory which allow us to formulate a systematic and controlled renormalization group analysis using a convenient dimensional regularization method. Moreover, this method fully preserves a two- dimensional Fermi surface with anisotropic dispersion in the vicinity of every point on the Fermi surface, and these features are crucial for our results. We will discuss the field theory for the Ising-nematic critical point, but similar field theories and results also apply to the problem of a Fermi surface coupled to a gauge field, or to other long-wavelength order parameters [28].

We consider a theory of fermions, ψ, in (2 + 1) dimensions which are coupled to a critical boson, Φ,

∫ ∫ ∑ ∑N 3 3 d k † 1 d k S(ψ,¯ ψ, Φ) = ψ˜ (k)(ik + sk + k2)ψ˜ (k) + (k2 + k2 + k2)Φ(−k)Φ(k) (2π)3 sj 0 x y sj 2 (2π)3 0 x y s= j=1 ∫ ∫ ∑ ∑N 3 3 e d k d q † + √ λ Φ(q)ψ˜ (k + q)ψ˜ (k), (2π)3 (2π)3 s sj sj N s= j=1

(4.11)

87 where e is the fermion-boson coupling constant, s = 1 labels the two Fermi surface patches, N is the number of fermionic flavors and λs equals 1 (s) for the Ising-nematic critical point (fermions coupled to a

U(1) gauge field). This model has been studied by many authors, including Refs. [28, 79]. In the following, we restrict ourselves to the Ising-nematic critical point and set λs = 1.

This model can be studied in a controlled way using the dimensional regularization scheme proposed by

Dalidovich and Lee [79]. Increasing the codimension of the Fermi surface by introducing auxiliary time-like directions, the dimensionally regularized action in (d + 1) dimensions reads

∫ ∫ ∑N dd+1k 1 dd+1q S(ψ,¯ ψ, Φ) = ψ¯ (k)[iΓ · K + iγ δ ]ψ (k) + [Q2 + q2 + q2]Φ(−q)Φ(q) (2π)d+1 j x k j 2 (2π)d+1 x y j=1 ∫ ∫ ie √ ∑N dd+1k dd+1q + √ d − 1 Φ(q)ψ¯ (k + q)γ ψ (k), (2π)d+1 (2π)d+1 j x j N j=1

(4.12)

where K = (k0, k1, . . . , kd−2) collects the physical and (d−2) auxiliary frequency variables. We introduced the spinor notation

( )T † † ψj(k) = ˜ ˜ − ψ¯j(k) = ψ (k)γ0 (4.13) ψ+,j(k), ψ−,j( k) j

and defined the gamma matrices as γ0 = σy and γx = σx for the spatial and as Γ = (γ0, γ1, . . . , γd−2) for the time-like directions. Within a patch, we choose kx (ky) perpendicular (parallel) to the Fermi surface, as shown in Fig. 4.1. The dispersion in the spatial plane containing the Fermi surface is δk, while the full dispersion is ε with k ( ) √ ∑d−2 1/2 − 2 2 2 δk = kx + d 1ky , εk = δk + ki . (4.14) i=1

Note the line of zero energy excitations in the plane ki = 0 which represents a patch on the Fermi surface in

Fig. 4.1, and the relativistic dispersion along the ki directions.

88 Rescaling momenta as

−1 ′ −1 ′ −1/2 ′ K = b K kx = b kx ky = b ky, (4.15)

∼ 2 the fermionic quadratic part of the action and the contribution qy in the bosonic quadratic part of the action are invariant under rescaling if fields are scaled as

d/2+3/4 ′ ′ d/2+3/4 ′ ′ ψj(k) = b ψj(k ) Φ(k) = b Φ (k ) (4.16)

2 2 The terms proportional to Q and qx in the bosonic quadratic part are irrelevant under this rescaling. The coupling scales as

′ 1 (5/2−d) e = eb 2 , (4.17) identifying d = 5/2 as the upper critical dimension. It is irrelevant for d > 5/2 and relevant for d < 5/2.

This allows to access non-Fermi liquid physics perturbatively by using ϵ = 5/2 − d as expansion parameter.

Keeping only marginal terms, the ansatz for the local field theory reads

∫ ∫ ∑N dd+1k 1 dd+1q S(ψ,¯ ψ, Φ) = ψ¯ (k)[iΓ · K + iγ δ ]ψ (k) + q2Φ(−q)Φ(q) (2π)d+1 j x k j 2 (2π)d+1 y j=1 ∫ ∫ (4.18) ieµϵ/2 √ ∑N dd+1k dd+1q + √ d − 1 Φ(q)ψ¯ (k + q)γ ψ (k), (2π)d+1 (2π)d+1 j x j N j=1 where we introduced the momentum scale µ in order to make the coupling e dimensionless. Perturbative corrections to this action at one-loop level reintroduce dynamics for the bosonic field. The ϵ = 5/2 − d expansion allows us to make a renormalized perturbative computation in the dimensionless coupling e. Note that this is not equivalent to a simple 1/N expansion, which breaks down at the Ising-nematic QCP [28], and

e4/3 that the expansion parameter is N [79].

89 (a) (b) (c)

Figure 4.2: Feynman diagrams yielding the renormalization of the scaling behavior of the viscosity at lowest order in ϵ: (a) One-loop contribution, (b) self-energy correction and (c) vertex correction. Lines represent fermionic propagators, wiggly lines bosonic propagators and curly lines the stress tensor.

4.4 Optical shear viscosity

In the following, we focus on the ‘optical’ shear viscosity, evaluated at frequencies ω ≫ T . Its evaluation is simpler than that for the d.c. viscosity, ω ≪ T , which will be considered in Section 4.5.

For the Ising-nematic QCP the xy-component of the stress tensor is proportional to the y-component of the ‘chiral’ current operator,

∫ ∑N ( ) qy 1 Txy(q) = i ky + ψ¯j(k + q)γxψj(k) = √ Jy(q). (4.19) 2 2 d − 1 j=1 k

∫ ∫ ∫ ∫ d−1 dkx dky d K where k = 2π 2π (2π)d−1 . Note that the x- and y-components of the chiral current contain the same gamma matrix.

The Feynman diagrams describing the renormalization of the scaling behavior of the viscosity at lowest order in ϵ are shown in Fig. 4.2.

At one-loop level, the stress tensor correlator is given by

∫ ( ) 2 ⟨Txy(q)Txy(−q)⟩1Loop = N (ky + qy/2) tr γxG0(k + q)γxG0(k) (4.20) k where

Γ · K + γxδk G0(k) = 2 2 (4.21) i(K + δk)

90 is the bare fermionic Green’s function. Specializing to q = ωe0,

∫ dd+1k δ2 − K · (K + Q) ⟨T T ⟩ (iω) = −2N k2 k ( ), (4.22) xy xy 1Loop d+1 y 2 2 2 2 (2π) (K + δk) (K + Q) + δk

where Q = ωe0. The further evaluation parallels that of the optical conductivity [161]. Shifting kx → √ − − 2 kx d 1ky eliminates ky from the integrand except in the prefactor arising from the stress tensor, yielding

∫ ∫ ∫ dk dk dd−1K k2 − K · (K + Q) ⟨T T ⟩ (iω) = −2N y k2 x x xy xy 1Loop 2π y 2π (2π)d−1 (K2 + k2)((K + Q)2 + k2) ∫ x x dk = −2N y k2I (Q). (4.23) 2π y 1loop

Introducing Feynman parameters, completing squares in the denominator and shifting K → K − (1 − x)Q, we obtain

∫ ∫ ∫ dd−1K dp 1 p2 − K2 + x(1 − x)Q2 I (Q) = dx 1loop (2π)d−1 (2π) [K2 + p2 + x(1 − x)Q2]2 ∫ ∫ 0 ∞ 1 2 d−1 2 πS − x(1 − x)Q S − √ Γ( )Γ(d/2) = d 1 dkkd−2 dx = d 1 πΓ(2 − d/2) 2 |Q|d−2 d 2 2 3/2 d (2π) 0 0 [k + x(1 − x)Q ] (2π) Γ(d) (4.24)

d/2 (Sd = 2π /Γ(d/2)). For d = 5/2 − ϵ, the one-loop result for the stress tensor autocorrelation function thus reads ∫ dk ⟨T T ⟩ (iω) = −Nu y k2|ω|1/2−ϵ, (4.25) xy xy 1loop 1Loop,ϵ 2π y where ϵ−1/2 3+2ϵ 5−2ϵ 2 2 Γ( 4 )Γ( 4 ) u1Loop,ϵ = √ . (4.26) 5/2−ϵ 5−2ϵ π Γ( 2 )

The momentum parallel to the Fermi surface, ky, does not scale due to the emergent rotational symmetry [28] of the low-energy field theory. The latter restricts the momentum dependence of the fermionic and bosonic

91 propagator to G(K, kx, ky) = G(K, δk) and D(Q, qx, qy) = D(Q, qy), respectively, which allows to eliminate ky from the integrand by shifting kx. The ky-integral is cut off by the Fermi surface curvature.

As a consequence, the result (4.25) differs from the current-current correlation function only by the fact that ∫ ∫ k2 appears instead of [161]. Importantly, both results have the same dependence on frequency. ky y ky

The two-loop self-energy correction to the optical viscosity is given by

∫ ( ) 2 ⟨TxyTxy⟩SE(iω) = 2N (ky + qy/2) tr γxG0(k + q)γxG0(k)Σ1(k)G0(k) , (4.27) k where ( ) e4/3 µ 2ϵ/3 Σ (k) = −i(Γ · Q) u ϵ−1 + O(ϵ0) (4.28) 1 N |Q| Σ,0

1/3 −1 is the one-loop fermionic self-energy [79](uΣ,0 = (2 · 6 π) ). We obtain

∫ ( ) 2ϵ/3 ⟨ ⟩ 4/3 −1 2| |1/2−ϵ µ TxyTxy SE(iω) = e ϵ ky ω | | aΣ,0, (4.29) ky ω

where aΣ,0 = u1Loop,0uΣ,0, after evaluation of the integrals as described in Appendix D.1. The dependence on frequency is the same as in the self-energy correction to the current-current correlation function [161].

The two-loop vertex correction is given by

∫ ( ) ⟨TxyTxy⟩VC(iω) = −iN (ky + qy/2) tr γxG0(k + q)Γxy,1(k, q)G0(k) , (4.30) k

where Γxy,1 is the one-loop correction to the stress-tensor. Ward identities due to the conservation of the chiral current imply that the vertex correction to the stress tensor correlation function does not have a pole in ϵ−1, as for the optical conductivity [79, 161]. At lowest order in ϵ, we thus obtain

∫ { ( ) } 4/3 2ϵ/3 ⟨ ⟩ − 2 | |1/2−ϵ − e µ TxyTxy (iω) = N kyu1Loop,0 ω 1 | | uΣ,0 + ... (4.31) ky Nϵ ω

92 for the correlator of the stress tensor. Evaluation of the coupling e4/3/N at the fixed point using the β function in O(ϵ) [79], ( 4/3 )∗ e − = u 1 ϵ, (4.32) N Σ,0

1/2−ϵ/3 and resummation of the frequency dependence yields ⟨TxyTxy⟩(iω) ∼ |ω| for the correlator and

η(ω) ∼ ω−1/2−ϵ/3 (4.33)

for the optical shear viscosity. Repeating the scaling arguments as described in Section 4.2 for two spatial dimensions, one time dimension and 1/2−ϵ auxiliary time dimensions, the optical shear viscosity is expected to scale as

− − η(ω) ∼ ω(d+(1/2 ϵ)z θη)/z, (4.34)

where θη is a hyperscaling violation exponent. The result in Eq. (4.33) corresponds to θη = 3, and thus

̸ 2 θη = θ. The origin of this breakdown of the scaling expectation is the ky factor in Eq. (4.31), which is dominated by contributions near the cutoff.

Instead, the result in Eq. (4.33) suggests that the viscosity scales like a conductivity. For the conductivity, the arguments in Section 4.2 imply that for the present dimensionally extended system, the scaling law in

Eq. (4.4) is modified to

σ(ω) ∼ ω(d+(1/2−ϵ)z−θ−2)/z; (4.35) using the values d = 2, θ = 1, and z = 3/(3 − 2ϵ), this agrees with Eq. (4.33). The Ward identity analysis in Appendix D.2 shows that the identity of the scaling between the viscosity and the conductivity holds to all orders.

In the above computation, we considered the contributions to the optical viscosity from two patches on the Fermi surface. In Appendix D.3, we show that the scaling is the same if contributions from the full Fermi

93 surface are taken into account. Moreover, by using Ward identities we trace the above conclusion back to the emergent rotation invariance of the low energy field theory, or equivalently to the fact that the Fermi surface curvature does not flow.

Given the scaling of entropy in the present system

s ∼ T (d+(1/2−ϵ)z−θ)/z, (4.36)

our main result in Eq. (4.2) would follow from Eq. (4.35) provided the viscosity scaled in the same manner with T in the regime ω ≪ T , as it does with ω in ω ≫ T . We will turn to this important question in the following section.

4.5 Boltzmann equation and DC viscosity

This section presents a Boltzmann equation analysis which shows that ω/T scaling applies, and that the ω- dependent results above can be extended to the d.c. viscosity with ω → T . We set N = 1 in this section for convenience. The DC viscosity may be derived in linear response by applying a static source that couples linearly to Txy, which is equivalent to applying a static source that couples linearly to Jy for the fermion contribution, i.e. a chiral electric field.

Since our action is invariant under inversion for the full Fermi surface, i.e.

ψ˜s(k0, kx, ky) = ψ˜s¯(k0, −kx, −ky), (4.37)

and this leaves Jy invariant but inverts the total momentum Pi → −Pi, the chiral current has zero overlap

94 with the conserved total momentum, i.e.

∫ 1/T ≡ ⟨ ⟩ χJyPi Jy(τ)Pi(0) = 0. (4.38) 0

Thus, the DC chiral conductivities and hence the DC viscosities are finite and can be determined using the Boltzmann equation. Fig. 4.3 illustrates how chiral currents can be excited without changing the total momentum of the system. This requires oppositely directed electric fields to be applied to the two patches.

The kinetic part of the fermion Hamiltonian in the dimensionally regularized theory may be diagonalized as ∫ ∫ ddk [ ] ∑ ddk H0 = ψ¯(k) iΓ¯ · K¯ + iγ δ ψ(k) = mλ† (k)ξ(k)λ (k), f (2π)d x k (2π)d m m m=

¯ where we use k = (kx, ky, K¯ ), K¯ = (k1, ..., kd−2) and Γ = (γ1, ..., γd−2), with the dispersion

( ) ¯ 2 2 1/2 ξ(k) = K + δk . (4.39)

The y-component of the chiral current density becomes

  ∫ ∑ d mδ ∂ δ  d k √ k ky k †  II Jy = d λm(k)λm(k) + Jy , (4.40) (2π) ¯ 2 2 m= K + δk

II † † where Jy contains particle-hole terms λ+λ−, λ−λ+ that are unimportant for transport in the DC regime of interest [107, 108]. Defining the non-equilibrium on-shell fermion distribution functions

m ⟨ † ⟩ ff (t, k) = λm(t, k)λm(t, k) , (4.41)

and the non-equilibrium off-shell boson distribution function fb(t, q, Ω), we can write down the following

95 collision equations in presence of an applied chiral electric field E [105, 107, 163]:

( ) ∫ ∑ d [ ] ∂ ∂ m 2 ϵ d q R ′ + E · f (p, t) = −e µ M ′ (p, q)Im D (p − q, mξ(p) − m ξ(q)) ∂t ∂p f (2π)d mm m′= { } × m − m′ − − ′ m − m′ ff (t, p)(1 ff (t, q)) + fb(t, p q, mξ(p) m ξ(q))(ff (t, p) ff (t, q)) , (4.42)

[ ] ∂ ∂ ∂Re[ΣR(t, q, Ω)] ∂ (2Ω2 − Re[ΣR(t, q, Ω)]) + b f (t, q, Ω) = ∂Ω b ∂t ∂t ∂Ω b ∫ [ ∑ d d k ′ ′ 2 ϵ ′ − − m − m 4πe µ d Mmm (k + q, k)δ(mξ(k + q) m ξ(k) Ω) ff (t, k + q)(1 ff (t, k)) ′ (2π) m,m = ] m − m′ + fb(t, q, Ω)(ff (t, k + q) ff (t, k)) , (4.43)

where the interaction matrix elements are

( ) 1 ′ δpδq − P¯ · Q¯ M ′ (p, q) = 1 + mm (4.44) mm 2 ξ(p)ξ(q)

(Note that M++ = M−−, M+− = M−+ and Mmm′ (p, q) = Mmm′ (q, p)), and

|k | DR( , ω) = y , k 3 2 ϵ 2 2 (d−1)/2 (4.45) |ky| + βde µ (K¯ − ω )

where βd depends only on d and is free of poles in ϵ [79]. The additional self-energy component appearing in the boson collision equation is given by [163]

∫ ′ ∑ d f m (t, + ) − f m(t, ) R 2 ϵ d k f k q f k Re[Σ (t, q, Ω)] = −2e µ M ′ (k, k + q) . (4.46) b (2π)d mm mξ(k) − m′ξ(k + q) + Ω mm′=

Both collision integrals vanish regardless of what DR is when the equilibrium distributions are used due to

96 the identity

nf (x)(1 − nf (y)) + nb(x − y)(nf (x) − nf (y)) = 0. (4.47)

We parameterize the deviations of the distributions from equilibrium in frequency space

m − ′ m ¯ · ∇ ff (ω, p) = 2πδ(ω)nf (mξ(p)) T nf (ξ(p))φ (ω, δp, P)E(ω) pξ(p),

fb(ω, q, Ω) = 2πδ(ω)nb(Ω) + u(ω, q, Ω)|E(ω)|. (4.48)

Using these, we linearize the collision equations with E = Eyeˆy as we are interested in Jy. In the DC limit,

+ m + we obtain (since limω→0(−iω + 0 )φ (ω, δp, P¯) and limω→0(−iω + 0 )u(ω, q, Ω) are expected to vanish in the presence of interactions)

√ ∫ − ∑ d [ ] 2mδp d 1py ′ 2 ϵ d q R ′ n (ξ(p)) = −e µ M ′ (p, q)Im D (p − q, mξ(p) − m ξ(q)) ξ(p) f (2π)d mm m′= { √ √ 2δ d − 1q ′ 2δ d − 1p × q y φm (δ , Q¯ )T n′ (ξ(q))n (mξ(p)) − p y φm(δ , P¯)T n′ (ξ(p))(1 − n (m′ξ(q))) ξ(q) q f f ξ(p) p f f ( √ √ ) 2δ d − 1q ′ 2δ d − 1p + n (mξ(p) − m′ξ(q)) q y φm (δ , Q¯ )T n′ (ξ(q)) − p y φm(δ , P¯)T n′ (ξ(p)) b ξ(q) q f ξ(p) p f } ′ ′ + sgn(Ey)u(p − q, mξ(p) − m ξ(q))(nf (mξ(p)) − nf (m ξ(q))) . (4.49)

where we have suppressed the now zero frequency argument on the φ’s and u’s. For the boson collision

97 equation we obtain

I1[φ, q, Ω] u(q, Ω) = sgn(Ey) , I2(q, Ω) ∫ ∑ d d k ′ 2 ϵ ′ − − I1[φ, q, Ω] = 4πe µ d Mmm (k + q, k)δ(mξ(k + q) m ξ(k) Ω) ′ (2π) [ mm = { √ 2δ d − 1k ′ × k y φm (δ , K¯ )T n′ (ξ(k))n (mξ(k + q)) ξ(k) k f f √ } 2δ d − 1(k + q ) − k+q y y m ¯ ¯ ′ − ′ φ (δk+q, K + Q)T nf (ξ(k + q))(1 nf (m ξ(k))) ξ(k + q) ] { √ √ } 2δ d − 1k ′ δ d − 1(k + q ) + n (Ω) k y φm (δ , K¯ )T n′ (ξ(k)) − 2 k+q y y φm(δ , K¯ + Q¯ )T n′ (ξ(k + q)) , b ξ(k) k f ξ(k + q) k+q f ∫ ∑ d 2 ϵ d k ′ I (q, Ω) = −4πe µ M ′ (k + q, k)δ(mξ(k + q) − m ξ(k) − Ω) 2 (2π)d mm mm′= { } ′ × nf (mξ(k + q)) − nf (m ξ(k)) . (4.50)

Since the driving term for the fermions in Eq. (4.43) is of opposite signs for the + and − quasiparticles,

m we expect φ (δp, P¯) = mφ(δp, P¯). Then, using the properties of the matrix elements M noted previously and that nf,b(x) + nf,b(−x) = 1, one can see that u is an odd function of Ω and hence that the same

φ(δk, K¯ ) can be used to solve the collision equations for both branches of quasiparticles. √ → − − 2 In the (convergent) boson collision integrals in Eq. (4.50), we shift kx kx d 1ky and integrate over ky. In the (also convergent) fermion collision integral Eq. (4.49), after inserting u derived from the √ → − − 2 → boson collision equation we shift qx qx d 1qy followed by qy qy + py, and then integrate out qy

98 √ after dividing through by 2 d − 1py. Terms that are odd in qy drop out, and we are left with

∫ ( ) mδ e2µϵ ∑ dd−1q δ q − P¯ · Q¯ p n′ (ξ(p)) = − 1 + mm′ p √x f d 2 2 ξ(p) 2 ′ (2π) ξ(p) Q¯ + q  m = x   ( ) − ( √ ) (d−1)/2 1/3  2  4π  2 ϵ 2 ′ 2 2  × Im √ βde µ (P¯ − Q¯ ) − mξ(p) − m Q¯ + q  27 x { (√ ) ′ qx m ′ 2 2 × √ φ (qx, Q¯ )T n Q¯ + q nf (mξ(p)) ¯ 2 2 f x Q + qx ( ( √ )) δ − p φm(δ , P¯)T n′ (ξ(p)) 1 − n m′ Q¯ 2 + q2 ξ(p) p f f x ( √ ) ( (√ ) ) ′ ′ 2 2 qx m ′ 2 2 δp m ′ + nb mξ(p) − m Q¯ + q √ φ (qx, Q¯ )T n Q¯ + q − φ (δp, P¯)T n (ξ(p)) x ¯ 2 2 f x ξ( ) f Q + qx p √ ( ( √ )) } − − ′ ¯ 2 2 H1[φ, P Q, mξ(p) m Q + qx] ′ 2 2 − √ nf (mξ(p)) − nf m Q¯ + q , (4.51) − | − ′ ¯ 2 2| x H2(P Q, mξ(p) m Q + qx ) where

∫ √ √ ∑ dd−1k |m′ K¯ 2 + k2 + Ω|Θ((m′ K¯ 2 + k2 + Ω)2 − (K¯ + Q¯ )2) H [φ, Q¯ , Ω] = ( x x ) 1 (2π)d √ 1/2 ′ ′ ¯ 2 2 2 − ¯ ¯ 2 mm s (m K + kx + Ω) (K + Q)  ( )  √ 1/2 ′ ¯ 2 2 2 − ¯ ¯ 2 − ¯ ¯ · ¯  s (m K + kx + Ω) (K + Q) kx (K + Q) K × 1 + mm′ √ √  | ′ ¯ 2 2 | ¯ 2 2 m K + kx + Ω K + kx [ s((m′(K¯ 2 + k2)1/2 + Ω)2 − (K¯ + Q¯ )2)1/2 × φm(s((m′(K¯ 2 + k2)1/2 + Ω)2 − (K¯ + Q¯ )2)1/2, K¯ + Q¯ ) x√ x | ′ ¯ 2 2 | m K + kx + Ω { } × ′ ′ ¯ 2 2 1/2 − ′ ¯ 2 2 1/2 T nf (m (K + kx) + Ω) 1 nf (m (K + kx) ) + nb(Ω) ] { } m′ kx ′ ′ 2 2 1/2 ′ 2 2 1/2 − φ (kx, K¯ )√ T n (m (K¯ + k ) ) nf (m (K¯ + k ) + Ω) + nb(Ω) , ¯ 2 2 f x x K + kx

99 ∫ √ √ ∑ dd−1k |m′ K¯ 2 + k2 + Ω|Θ((m′ K¯ 2 + k2 + Ω)2 − (K¯ + Q¯ )2) H (Q¯ , Ω) = ( x x ) 2 (2π)d √ 1/2 ′ ′ ¯ 2 2 2 − ¯ ¯ 2 mm s (m K + kx + Ω) (K + Q)  ( )  √ 1/2 ′ ¯ 2 2 2 − ¯ ¯ 2 − ¯ ¯ · ¯  s (m K + kx + Ω) (K + Q) kx (K + Q) K × 1 + mm′ √ √  | ′ ¯ 2 2 | ¯ 2 2 m K + kx + Ω K + kx { ( √ ) ( √ )} × ′ ¯ 2 2 − ′ ¯ 2 2 nf m K + kx + Ω nf m K + kx . (4.52)

m ¯ ¯ ¯ 2 ¯2 1/2 If we choose φ (δp, P) = φ(δp, P) with φ(δp, P) = C(T )(δp+P ) /δp, the right hand side of Eq. (4.51) vanishes due to the identity

′ − ′ − − ′ − ′ nf (x)(nf (y) 1) + nf (x)nf (y) nb(x y)(nf (x) nf (y)) = 0. (4.53)

This is the zero mode of the collision equation, and will lead to an infinite conductivity if excited. However, this mode cannot be excited by the chiral electric field as it produces the same (instead of opposite) deviation in the + and − quasiparticle distributions. This mode will be excited by a normal electric field, and is responsible for the infinite DC charge conductivity of the system. The modes excited by the chiral electric

m field obey φ (δp, P¯) = mφ(δp, P¯) and are orthogonal to the zero mode, yielding a finite chiral conductivity

(or viscosity).

We have

∫ ∫ J dd−1p p2 η ∼ σ = y = 8(1 − d)T dp p2 x n′ ((P¯2 + p2)1/2)φ(p , P¯), (4.54) yy y y d ¯2 2 f x x Ey (2π) P + px

√ → − − 2 where we shifted px px d 1py. Counting powers in Eq. (4.51), we obtain

¯ 1/3 −4/3 −2ϵ/3 −2(d−1)/3−1 ¯ φ(ξ(p), P) = βd e µ T φ˜(ξ(p)/T, P/T ). (4.55)

Inserting this into Eq. (4.54) and using the fixed point values of z∗ = 3/(2(d − 1)) and e∗4/3 ∝ ϵ [79], we

100 Figure 4.3: Elementary excitations due to the chiral electric field that carry a net chiral current at zero total momentum relative to the filled band in a two-patch system. The chiral current can decay via the emission of bosons of opposite momenta on the two patches. Since the individual bosons carry nonzero momentum, the boson distribution responds to the applied chiral electric field and is no longer in equilibrium unlike in a particle-hole symmetric system, where the bosons required to relax the elementary excitations have zero momentum. have, to leading order in ϵ, ∫ 1 η ∼ T −1/z+d−2 dp p2, (4.56) ϵ y y which is the expected quantum critical scaling.

If the Boltzmann analysis at this order is performed directly in d = 2, then the collision equations are solved exactly by using the collisionless momentum-independent solution for φ, and thus collisions with the boson do not induce a finite DC viscosity. The reason for this is purely kinematic, stemming from the special structure of the patch dispersions in d = 2 which have Galilean invariance in the y direction and a constant x velocity and was noted earlier in Ref. [128]. The quantum critical scaling could possibly be restored by appropriately resumming contributions at higher orders in perturbation theory.

101 4.6 Discussion this chapter has exposed the unconventional scaling of the shear viscosity in a theory with a critical Fermi surface. For the Ising-nematic QCP in d = 2, we computed the optical and DC viscosities in an expansion in

ϵ = 5/2−d below the upper critical dimension, and showed that the viscosity scales differently than expected from that of a critical point with an effective reduced dimensionality of (d − 1)-dimensional excitations transverse to the Fermi surface. As a consequence, the ratio η/s diverges at low temperatures as T −2/z instead of saturating the universal bound like in other strongly-coupled field theories in the literature. We expect that this is a general phenomenon of metallic quantum critical states where hyperscaling is violated due to the presence of a critical Fermi surface, including states described by Fermi surfaces coupled to gauge fields. However, we do expect that metallic critical points associated with singular ‘hot spots’ on the Fermi surface [163] will have a finite η/s, up to logarithmic factors.

102 Strangeness is a necessary ingredient in beauty.

Charles Baudelaire

5 Magnetotransport in a model of a disordered

strange metal

5.1 Introduction

Essentially all correlated electron high temperature superconductors display an anomalous metallic state at temperatures above the superconducting critical temperature at optimal doping [6–8]. This metallic state has a ‘strange’ linearly-increasing dependence of the resistivity, ρ, on temperature, T ; it can also exhibit bad metal behavior with a resistivity much larger than the quantum unit ρ ≫ h/e2 (in two spatial dimensions)

[216]. More recently, strange metals have also been demonstrated to have a remarkable linear-in-B magne-

103 toresistance, with the crossover between the linear-in-T and linear-in-B behavior occurring at µBB ∼ kBT

[17, 19].

this chapter will present a model of a strange metal which exhibits the above linear-in-T and linear-in-B behavior. The model builds on a lattice array of quantum ‘dots’ or ‘islands’, each of which is described by a

Sachdev-Ye-Kitaev (SYK) model of fermions with random all-to-all interactions [35, 36]. The SYK models are 0+1 dimensional quantum theories which exhibit a ‘local criticality’. They have drawn a great deal of interest for a variety of reasons:

(i) The SYK models are the simplest solvable models without quasiparticle excitations. They can also be used as fully quantum building blocks for theories of strange metals in non-zero spatial dimensions [42, 217].

(ii) The SYK models exhibit many-body chaos [36, 218], and saturate the lower bound on the Lyapunov time to reach chaos [46]. So they are “the most chaotic” quantum many-body systems. The presence of maximal chaos is linked to the absence of quasiparticle excitations, and the proposed [53] lower bound of order ℏ/(kBT ) on a ‘dephasing time’. It is important to note here that the co-existence of many-body chaos and solvability is quite remarkable: essentially all other solvable models (e.g. integrable lattice models in one dimension) do not exhibit many-body chaos.

(iii) Related to their chaos, the SYK models exhibit [219] eigenstate thermalization (ETH) [220, 221], and yet many aspects are exactly solvable.

(iv) The SYK models are dual to gravitational theories in 1 + 1 dimensions which have a black hole horizon. The connection between the SYK models and black holes with a near-horizon AdS2 geometry was proposed in Refs. [222, 223], and made much sharper in Refs. [36, 224, 225]. This connection has been used to examine aspects of the black hole information problem [226].

More specifically, a single SYK site is a 0+1 dimensional non-Fermi liquid in which the imaginary-time

104 (τ) fermion Green’s function has the low T ‘conformal’ form [35, 37, 217, 227]

( ) T 1/2 G(τ) ∼ e−2πET τ , 0 < τ < 1/T , (5.1) sin(πT τ) where E is a parameter controlling the particle-hole asymmetry. In frequency space, this correlator is √ G(ω) ∼ 1/ ω for ω ≫ T , and this implies non-Fermi liquid behavior. A Fermi liquid has the exponent 1/2 in Eq. (5.1) replaced by unity, and a constant density of states with G(ω) frequency independent.

The Green’s function in Eq. (5.1) implies [35] a ‘marginal’ [228] susceptibility, χ, with a real part which diverges logarithmically with vanishing frequency (ω) or T . Specifically, in the all-to-all limit of the SYK model, vertex corrections are sub-dominant, and Fourier transform of χ(τ) = −G(τ)G(−τ) leads to the spectral density ( ) ω Im χ(ω) ∼ tanh , (5.2) 2T whose Hilbert transform leads to the noted logarithmic divergence. In contrast, a Fermi liquid has Im χ(ω) ∼

ω. The form in Eq. (5.2) is consistent with recent electron scattering observations [229]. A linear-in-T resistivity now follows upon considering itinerant fermions scattering off such a local susceptibility, and the itinerant fermions realize a marginal Fermi liquid (MFL) with a ω ln ω self energy [35, 222, 228, 230].

We now review previous approaches to building a finite-dimensional non-Fermi liquid from the 0 + 1 dimensional SYK model. An early model for a bulk strange metal in finite spatial dimensions was provided by Parcollet and Georges [217]. They considered a doped Mott insulator described by a random t-J model at hole density δ, where t is the root-mean-square (r.m.s.) electron hopping, and J is the r.m.s. exchange interaction. At low doping with δt ≪ J, they found strange metal behavior in the intermediate T regime

2 Ec < T < J, where the coherence energy Ec = (δt) /J. In this intermediate energy range, they found that the electron Green’s function had the local form of the SYK model in Eq. (5.1). Moreover, this metal

2 2 had ‘bad metal’ resistivity with ρ ∼ (h/e )(T/Ec) ≫ (h/e ). We will refer to such a strange metal as an

105 ‘incoherent metal’ (IM). This IM is to be contrasted from a MFL, which we will describe below; the MFL does not appear in the model of Parcollet and Georges.

Another finite-dimensional model of an IM appeared in the recent work of Song et al. [42]. They considered a lattice of SYK sites, with r.m.s. on-site interaction U, and r.m.s. inter-site hopping t. Each site was a quantum island with N orbitals, and had random on-site interactions with typical magnitude U. Elec- trons were allowed to hop between nearest-neighbor states, with a random matrix element of magnitude t. Although this is a model with strong interactions, the remarkable fact is that the random nature of the interactions renders it exactly solvable. As in Ref. [217], Song et al. found an IM in the intermediate regime Ec < T < U, with a local electron Green’s function as in Eq. (5.1), and a bad metal resistivity

2 2 ρ ∼ (h/e )(T/Ec). Their coherence scale was Ec = t /U. (This lattice SYK model should be contrasted from earlier studies [40, 41], which only had fermion interaction terms between neighboring SYK sites: the latter models realize disordered metallic states without quasiparticle excitations as T → 0, but have a

T -independent resistivity.)

Although these models [42, 217] reproduce bad metal resistivity, we will show here that they are unable to describe the experimentally observed large magnetoresistance noted earlier [17, 19]. The random nature of the hopping between the sites, and the associated absence of a Fermi surface, results in negligible magnetoresistance. Significant orbital magnetoresistance only appears in models which have fermions with non-random hopping and a well-defined Fermi surface. Note that the existence of a Fermi surface does not directly imply the presence of well-defined quasiparticles: it is possible to have a sharp Fermi surface in momentum space (where the inverse fermion Green’s function vanishes) while the quasiparticle spectral function is broad in frequency space.

With the aim of obtaining a well-defined Fermi surface of itinerant electrons, in this chapter we consider a lattice of SYK islands coupled to a separate band of itinerant conduction electrons as illustrated in

Fig. 5.1. Our model is in the spirit of effective Kondo lattice models which have been proposed as models

106 Figure 5.1: (a) A cartoon of our microscopic model. Itinerant conduction electrons (green) hop around on a lattice (black). At each lattice site, they interact locally and randomly with SYK quantum dots (blue) through an interaction (orange) that independently conserves the numbers of conduction and island electrons. (b) Finite-temperature regimes of the model. When the conduction electron bandwidth is large enough, it realizes a disordered marginal-Fermi liquid (MFL) for the conduction electrons for all temperatures T ≪ J (Sec. 5.3.1). For a finite bandwidth, there can be a finite-temperature crossover to an ‘incoherent metal’ (IM), in which all notion of electron momentum is lost, if the coupling g is large enough (Sec. 5.3.2). Note that we always have J ≫ T and J ≳ g. of the physics of the disordered, single-band Hubbard model [231–233]. Other two band models of itinerant electrons coupled to SYK excitations have been considered in Refs. [234, 235]. Our model exhibits MFL behavior as T → 0, with a linear-in-T resistivity, and a T ln T specific heat. For an appropriate range of parameters, there is a crossover at higher T to an IM regime, also with a linear-in-T resistivity. The itinerant electrons have a non-random hopping t, the SYK sites have a random interaction with r.m.s. strength J, and these two sub-systems interact with a random Kondo-like exchange of r.m.s. strength g: see Fig. 5.1a for a schematic illustration. Fig. 5.1b illustrates the regimes of MFL and IM behavior in our model. In the MFL regime, our model exhibits a well-defined Fermi surface, albeit of damped quasiparticles.

The magnetotransport properties of this model will be a significant focus of our analysis. We will show that the MFL regime with a Fermi surface indeed has a sizeable magnetoresistance, with characteristics in accord with observations. We find that the longitudinal and Hall conductivities, of the MFL regime, can be written as scaling functions of B/T , as shown in Eq. (5.39). In contrast, the B dependence is much less singular in the

IM regime. Although a B/T scaling is obtained in the MFL in this computation, the magnetoresistance does not increase linearly with B, and instead saturates at large B. To obtain a non-saturating magnetoresistance

107 we consider a macroscopically disordered sample with domains of MFLs with varying electron densities; employing earlier work on classical electrical transport in inhomogeneous ohmic conductors [236–242], we obtain the observed linear-in-B magnetoresistance with a crossover scale at B ∼ T .

this chapter is organized as follows: In Sec. 5.2, we introduce our basic microscopic model of a disordered

MFL, and determine its single-electron properties and finite-temperature crossovers in Sec. 5.3. In Sec. 5.4, we solve for transport and magnetotransport properties of this basic model exactly in various analytically- tractable regimes. In Sec. 5.5, we introduce the effective-medium approximation and apply it to a macroscopically disordered sample containing domains of the basic model, obtaining analytical results for the global magnetotransport properties for certain simplified considerations of macroscopic disorder. We summarize our results and place them in the context of recent experiments in Sec. 5.6.

5.2 Microscopic model

We consider M flavors of conduction electrons, c, hopping on a lattice that are coupled locally and randomly to SYK islands on each lattice site (Fig. 5.1a). The islands contain N flavors of valence electrons, f, which interact among themselves in such a way that they realize SYK models. The Hamiltonian for our system is given by

∑M ∑M ∑N † † † − ′ − − H = t (cricr i + h.c.) µc cricri µ frifri ⟨rr′⟩; i=1 r; i=1 r; i=1 ∑N ∑M ∑N 1 † † 1 † † + gr f f c c + J r f f f f . (5.3) NM 1/2 ijkl ri rj rk rl N 3/2 ijkl ri rj rk rl r; i,j=1 k,l=1 r; i,j,k,l=1

We will take the limits of M = ∞ and N = ∞, but we will be interested in values of M/N that are at most

O r r ≪ r r′ ≫ (1). We choose Jijkl and gijkl as independent complex Gaussian random variables, with JijklJlkij =

′ 2 ′ ≪ r r ≫ 2 ′ ≪ ≫ ≪ ≫ (J /8)δrr and gijklgjilk = g δrr and all other .. ’s being zero, where .. denotes disorder-

108 averaging. Note that t is non-random, and this will lead to a Fermi surface for the c fermions. The disorder- averaged action then is

  ∫ β ∑M ∑M ∑N † † † ′  − − ′ −  S = dτ cri(τ)(∂τ µc)cri(τ) t (cri(τ)cr i(τ) + h.c.) + fri(τ)(∂τ µ)fri(τ ) 0 r; i=1 ⟨rr′⟩; i=1 r; i=1 ∫ g2 ∑ β − M dτdτ ′Gc(τ − τ ′)Gc(τ ′ − τ)G (τ − τ ′)G (τ ′ − τ) 2 r r r r r 0 ∫ J 2 ∑ β − N dτdτ ′G2(τ − τ ′)G2(τ ′ − τ) 4 r r r 0 ∫ ( ) ∑ β ∑N 1 † − N dτdτ ′Σ (τ − τ ′) G (τ ′ − τ) + f (τ)f (τ ′) r r N ri ri r 0 i=1 ∫ ( ) ∑ β ∑M 1 † − M dτdτ ′Σc(τ − τ ′) Gc(τ ′ − τ) + c (τ)c (τ ′) , (5.4) r r M ri ri r 0 i=1 where we have followed the usual strategy for SYK models [37, 41] and introduced the auxiliary fields

G, Σ,Gc, Σc corresponding to Green’s functions and self-energies of the f and c fermions respectively at each lattice site. In the M,N = ∞ limit, the integrals over the Σ, Σc fields enforce the definitions of G, Gc at each lattice site r. The large M, N saddle-point equations are obtained by varying the action with respect to these G and Σ fields after integrating out the fermions

M Σ (τ − τ ′) = Σ(τ − τ ′) = −J 2G2(τ − τ ′)G (τ ′ − τ) − g2G (τ − τ ′)Gc(τ − τ ′)Gc(τ ′ − τ) r r r N r r r M = −J 2G2(τ − τ ′)G(τ ′ − τ) − g2G(τ − τ ′)Gc(τ − τ ′)Gc(τ ′ − τ), N 1 G(iωn) = , (5.5) iωn + µ − Σ(iωn) and

c − ′ c − ′ − 2 c − ′ − ′ ′ − − 2 c − ′ − ′ ′ − Σr(τ τ ) = Σ (τ τ ) = g Gr(τ τ )Gr(τ τ )Gr(τ τ) = g G (τ τ )G(τ τ )G(τ τ), ∫ ∫ d d c d k 1 ≡ d k c G (iωn) = d c d G (k, iωn). (5.6) (2π) iωn − ϵk + µc − Σ (iωn) (2π)

109 The last expression shows that the c fermions have a dispersion ϵk and an associated Fermi surface; the lifetime of the Fermi surface excitations will be determined by the frequency dependence of Σc, which will be computed in the next section. We define chemical potentials such that half-filling occurs when µ = µc = 0.

The islands are not capable of exchanging electrons with the Fermi sea, so there is no reason a priori to have

µ = µc, or even for islands at different sites to have the same µ. However, for convenience we will keep the µ of all the islands the same. The real system would operate at fixed densities, and µ and µc will appropriately renormalize as the mutual coupling g is varied, in order to keep the densities of c and f individually fixed, as the interaction between c and f conserves their numbers individually. However, as we shall find, the half-filled case always corresponds to µ = µc = 0 regardless of g. We will always have J ≫ T in this chapter, and also J ≳ g. A sketch of the phases realized by our model as a function of temperature is shown in Fig. 5.1b.

5.3 Fate of the conduction electrons

5.3.1 The case of infinite bandwidth

We first consider the case of infinite bandwidth, or equivalently t ≫ g, J ≫ T . The precise value of µc doesn’t matter as long as its magnitude is not infinite, as the conduction electrons float on an effectively infinitely deep Fermi sea. Then, we can use the standard trick for evaluating integrals about a Fermi surface, and we have

∫ ∫ d ∞ c d k 1 → dε 1 G (iωn) = d c ν(0) c , (5.7) (2π) iωn − ϵk + µc − Σ (iωn) −∞ 2π iωn − ε − Σ (iωn) where ν(0) is the density of states at the Fermi energy.

We take the lattice constant a to be 1. This makes k dimensionless by redefining ka to be k. The energy

110 dimension of ϵk then comes from the inverse band mass. The density of states ν(0) then has the dimension of 1/(energy) (on a lattice ν(0) ∼ 1/t ∼ 1/Λ, where Λ is the bandwidth).

c We will also have sgn(Im[Σ (iωn)]) = −sgn(ωn), so

i ν(0)T Gc(iω ) = − ν(0)sgn(ω ),Gc(τ) = − , − β ≤ τ ≤ β, (5.8) n 2 n 2 sin(πT τ) with other intervals obtained by applying the Kubo-Martin-Schwinger (KMS) condition Gc(τ + β) =

−Gc(τ). At T = 0, we have ν(0) Gc(τ, T = 0) = − . (5.9) 2πτ

We consider M/N = 0 to begin with. Then, the f electrons are not affected by the c electrons, and their

Green’s functions are exactly of the incoherent form of the SYK model, which, in the low-energy limit, are given by [37]

( ) π1/4 cosh1/4(2πE) T 1/2 G(τ) = − √ e−2πET τ , 0 ≤ τ < β (5.10) J 1/2 1 + e−4πE sin(πT τ) where E is a function of µ with E ∝ −µ/J for small µ/J. Other intervals are again obtained by the KMS condition G(τ + β) = −G(τ). The zero-temperature limit of this, and similar expressions appearing later, can be straightforwardly taken [37]

cosh1/4(2πE) 1 cosh1/4(2πE) 1 G(τ > 0,T = 0) = − √ ,G(τ < 0,T = 0) = √ π1/4J 1/2 1 + e−4πE τ 1/2 π1/4J 1/2 1 + e4πE |τ|1/2 (5.11)

Now we can compute the self energy of the c fermions, which is

π1/2g2ν(0)T 2 Σc(τ) = −g2Gc(τ)G(τ)G(−τ) = − , 0 ≤ τ < β. (5.12) 4J cosh1/2(2πE) sin2(πT τ)

111 Fourier transforming with a cutoff of τ at J −1 ≪ T −1 and β − J −1 gives

( ( ) ( ) ) 2 γE −1 c ig ν(0)T ωn 2πT e ωn ωn Σ (iωn) = ln + ψ + π , (5.13) 2J cosh1/2(2πE)π3/2 T J T 2πT

where ψ is the digamma function and γE is the Euler-Mascheroni constant. As foreseen, this satisfies

c sgn(Im[Σ (iωn)]) = −sgn(ωn) on the fermionic Matsubara frequencies. For |ωn| ≫ T

( ) − 2 | | γE 1 c ig ν(0) ωn e Σ (iωn) → ωn ln . (5.14) 2J cosh1/2(2πE)π3/2 J

Note the MFL form of the itinerant c fermion self energy, ∼ ω ln ω. Since the large N and M limits are taken at the outset, this MFL is stable even as T → 0. For finite N and M, the coupling g is irrelevant in the infrared (IR) [235], and the model reduces to a theory of non-interacting electrons as T → 0, with the

MFL existing only above a temperature scale whose magnitude is suppressed in N and the zero-temperature entropy going to zero.

+ Upon analytically continuing iωn → ω + i0 , we get the inverse lifetime for the conduction electrons defined by 2 ≡ − c ≡ − c → + g ν(0)T γ 2Im[ΣR(0)] Im[Σ (iωn 0 + i0 )] = . (5.15) J cosh1/2(2πE)π1/2

Since the coupling of the conduction electrons to the SYK islands is spatially disordered, this rate also represents the transport scattering rate up to a constant numerical factor. The scattering of c electrons off the islands requires the f electrons inside the islands to move between orbitals. Hence γ vanishes when the islands are flooded or drained by sending E → ∓∞ respectively, say, by doping them.

If we do not have M/N = 0, the SYK Green’s function will be affected as there is a back-reaction self- energy to the SYK islands. To see what this does when we perturbatively turn on M/N, we compute it with

112 the M/N = 0 Green’s functions with a cutoff of τ at J −1 and β − J −1

M Mπ1/4 cosh1/4(2πE)g2ν2(0)T 5/2e−2πET τ Σ(˜ τ) = − g2G(τ)Gc(τ)Gc(−τ) ≈ − √ . (5.16) N 4NJ 1/2 1 + e−4πE sin5/2(πT τ)

˜ 2 2 If E = 0, then Σ(iωn) ∝ i(M/N)g ν (0)ωn as T, ωn → 0, which is sub-leading to Σ(iωn)|M/N=0 ∼

1/2 (Jωn) , so the SYK character of the islands survives in the IR.

Now we consider the case of particle-hole symmetry breaking with a non-zero spectral asymmetry, E in Eq. (5.1); we will find that the basic structure of the results described above persists. If E ̸= 0 but is

2 2 2 2 small, then for T → 0, Σ(˜ iωn → 0) ∼ −(M/N)g ν (0)JE ∝ (M/N)g ν (0)µ + O(iωn). In contrast

1/2 ˜ Σ(iωn → 0)|M/N=0 ∼ µ+O(ωn ). Therefore the frequency-dependent part of Σ is still subleading. Hence, in the IR we may still assume that all that happens to the SYK islands is that their chemical potential µ gets renormalized. By solving Re[Σ(iωn → 0,T = 0)] = µ, we obtain the corrected E ↔ µ relation. At small

µ/J, this is µ/J E ≈ − ( ). (5.17) √ g2ν2(0)M π1/4 2 1 + 6π3/2N ∑ N † The total particle number on each island, r = i firfir, commutes with H. Since the SYK particle density

Q = N /N is a universal function of E, independent of µ and J,(5.17) just implies a renormalization of the nonuniversal UV parts of the SYK Green’s function and the island chemical potential, while the particle density remains fixed. Similarly, the vanishing of the zero-frequency real part of (5.13) regardless of E implies that there is no renormalization of either the density or chemical potential of the conduction electrons in this infinite-bandwidth limit, since their number is independently conserved as well. For a finite bandwidth, the chemical potential of the conduction electrons renormalizes in such a way that their density remains fixed.

113 In Appendix E.1, we consider the effects of adding a ‘pair-hopping’ term to (5.3),

∑N ∑M [ ] 1 † † H → H + ηr f f c c + h.c. , (5.18) NM 1/2 ijkl ri rj rk rl r; i,j=1 k,l=1

≪ | r |2 ≫ 2 ≳ † † with ηijkl = η /8, and J η. This term has identical power-counting to the f fc c term, but can trade c electrons for f electrons and vice-versa. Since the numbers of c and f electrons are no longer independently conserved in this case, there is only one chemical potential, and µc = µ. We find that this term also generates an MFL as long as the bandwidth of the c electrons is large.

As is well known, the marginal-Fermi liquid self-energy we obtained (5.13, 5.14) also leads to the leading

MFL ∼ low-temperature contribution to the specific heat coming from the itinerant electrons scaling as CV

Mg2(ν(0))2(T/J) ln(J/T ) [243]. Note that the entropy has a non-vanishing T → 0 limit from the contribution of the SYK islands in the limit of N → ∞ [244], but this does not contribute to the specific heat. The contribution to the specific heat coming from the SYK islands scales linearly in T as T → 0 [41], which is subleading to the T ln T contribution of the itinerant electrons.

5.3.2 The case of a finite bandwidth

This subsection will show that a finite bandwidth does not modify the basic structure of the low-temperature

MFL phase described above. However, if interactions between c and f are strong enough, a crossover into an IM phase is possible at higher temperatures. Readers not interested in the details of the arguments can move ahead to the next section.

If the bandwidth (and hence Fermi energy) of the conduction electrons is sizeable compared to the cou-

c plings, then the momentum-integrated local Green’s function G (iωn) is no longer independent of the de-

c tails of the self energy Σ (iωn). We consider two spatial dimensions, with the isotropic dispersion εk =

2 − max − k /(2m) Λ/2, and a bandwidth εk εk=0 = Λ. Since k is dimensionless, the band mass m has dimen-

114 sions of 1/(energy). The density of states is then just ν(ε) = ν(0) = m, at all energies ε, and we implicitly make use of this fact while simplifying and rewriting certain expressions. On a lattice, m ∼ ν(0) ∼ 1/t ∼

1/Λ.

The momentum-integrated conduction electron Green’s function is

ν(0) Gc(iω ) = [ln(Λ + 2µ + 2iω − 2Σc(iω )) − ln(2µ − Λ + 2iω − 2Σc(iω ))] . (5.19) n 2π c n n c n n

c We still expect sgn(Im[Σ (iωn)]) = −sgn(ωn). The chemical potential µc must now take an appropriate value to reproduce the correct density of conduction electrons. The conduction band filling is given by

2πGc(τ = 0−) Q = , (5.20) c ν(0)Λ for the exact solution to Gc, which can be found by the imaginary-time MATLAB code ggc.m [245] (The low- energy ‘conformal-limit’ solutions described below are not valid at the short times 0−, and do not display this property).

In general, the Dyson equations can now only be solved numerically, which the imaginary-time MATLAB code ggc.m [245] and real-time MATLAB code ggcrealtime.m [246] do, albeit by holding the chemical

c potentials µ and µc, rather than densities, fixed. In an extreme limit where |iωn + µc − Σ (iωn)| far exceeds the bandwidth for all ωn, which can happen only at T ≠ 0, we have a simplification of (5.19), obtained by expanding in Λ,

c Λν(0) G (iωn) = c . (5.21) 2π(iωn + µc − Σ (iωn))

This then leads to an SYK solution in the low-energy conformal limit for both G and Gc, realizing a fully

115 incoherent metal. We use the trial solutions

( ) 1/2 C T − E c −√ c 2π cT τ G (τ) = − E e , 1 + e 4π c sin(πT τ) ( ) C T 1/2 G(τ) = −√ e−2πET τ , 0 ≤ τ < β. (5.22) 1 + e−4πE sin(πT τ)

Ec is universally related to the conduction band filling, with Ec = 0 at half filling, and Ec → ∓∞ when the band is full or empty respectively. When M/N = 0, there is no back-reaction to the islands, and G

c c is given by (5.10). We use the conditions Re[Σ (iωn → 0,T = 0)] = µc and G (iωn → 0,T = 0) =

c Λν(0)/(2π(µc − Σ (iωn → 0,T = 0))) to determine Cc, and also µc in terms of the fixed Ec. Cutting off

−1 τ integrals in the Fourier transforms at a distance αUV from singularities, we have

√ cosh1/4(2πE) g2 π1/4 cosh1/4(2πE)µ J C = ,J ≡ E ≈ − c ( µ /g), c 1/2 IM and c 1/2 1/2 At small c (5.23) 1/2 1/4 JΛν(0) gΛ ν (0) αUV 2 π JIM

c with no feedback on the SYK islands. For (5.21) to derive from (5.19), this requires |µc −Σ (iωn → 0)| ≫ Λ or ΛJ T ≫ T ≡ . (5.24) inc ν(0)g2

2 Furthermore, for (5.10) and (5.22) to hold, we also need J ≫ Tinc and JIM ≫ Tinc, implying g ≫ ΛJ.

For T ≪ Tinc, we go back to the MFL, which now has a UV cutoff of Tinc instead of J, with its self energy

c 2 going as Σ (iωn) ∼ (g ν(0)/J)iωn ln(|ωn|/Tinc). The choice of the UV cutoff αUV in the IM only affects the nonuniversal Ec ↔ µc relation. An appropriate choice of the cutoff is αUV ∼ JIM ≲ J.

Turning on a small but finite M/N, we have to additionally use the conditions Re[Σ(iωn → 0,T = 0)] =

µ and G(iωn → 0,T = 0) = 1/(µ − Σ(iωn → 0,T = 0)) simultaneously to determine a renormalized C and renormalized µ, while keeping E fixed as before. We again cut off τ integrals in the Fourier transforms

116 −1 at a distance αUV from singularities. This gives

( ) π1/4 M Λν(0) cosh(2πE) 1/4 cosh1/2(2πE)Λ1/2ν1/2(0) C = 1/4(2πE) 1 − ,C = , cosh 1/2 c 1/2 (5.25) J N 2π cosh(2πEc) 2 Cg

and we do not show the nonuniversal E, Ec ↔ µ, µc relations because they are rather uninsightful and the physics is better described in terms of E, Ec which universally represent the conserved densities.

If M/N is increased to approach (2π cosh(2πEc))/(Λν(0) cosh(2πE)), the condition for incoherence that

c |iωn + µc − Σ (iωn)| exceed the bandwidth for all ωn becomes harder to fulfill, and larger and larger values of the coupling g are required to achieve the IM phase at high temperatures.

When M/N > (2π cosh(2πEc))/(Λν(0) cosh(2πE)), we still recover the MFL deep enough in the IR, due to the back-reaction self energy Σ˜ being irrelevant, and the conduction electron self energy Σc also vanishing at the lowest energies. However, at values of the coupling g large enough so that effects of the conduction electron bandwidth may be ignored above a certain temperature, we find a crossover into a different IM phase, with local Green’s functions given by (at half-filling)

( ) ( ) − T ∆c T 1 ∆c Gc(τ) ∼ ,G(τ) ∼ , 0 < ∆ < 1/2, (5.26) sin(πT τ) sin(πT τ) c

with ∆c given by the solution to the equation

( ) ( ) ∆ π∆ M Λν(0) c cot2 c = , (5.27) 1 − ∆c 2 N 2π

which has the property that ∆c → 0 as M/N → ∞ and ∆c → 1/2 as M/N → 2π/(Λν(0)). These Green’s functions may be derived by solving the Dyson equations (5.5, 5.6) while ignoring both the conduction electron dispersion and the coupling J. Indeed, with the scalings in (5.26), the term proportional to J 2 in the expression for Σ(τ) is irrelevant compared to the other term. This phase has a resistivity that scales as

117 − T 2(1 ∆c). Since we are only interested in models with linear-in-T resistivities, we will henceforth assume that M/N is small enough to avoid this regime.

Since ν(0) ∼ 1/Λ ∼ 1/t on a lattice, fine-tuning g ∼ J ∼ Λ ≫ T makes the scattering rate (5.15)

‘Planckian’, i.e. an O(1) number times T , since it is given by ratios of large quantities. The MFL doesn’t

c break down if we do this; In (5.19), |Σ (i(ωn ∼ T ))| ∼ T ln T/J ≪ Λ, so the infinite-bandwidth result (5.15) is still applicable. The crossover to the IM doesn’t occur either, since T ≪ Tinc, and finally, the part of the back-reaction self-energy to the SYK islands that does not renormalize their chemical potentials is

2 1/2 |Σ(˜ i(ωn ∼ T ))]| ∼ (M/N)(gν(0)) T which is ≪ |Σ(i(ωn ∼ T ))| ∼ (JT ) , i.e. the part of the internal self-energy of the SYK islands that doesn’t renormalize chemical potential, as long as M/N is not ≫ 1, so the SYK character of the islands also survives.

In the IM regime, since both the conduction and island electrons have local SYK Green’s functions, the

IM ∼ specific heat scales as CV MT/JIM + NT/J, with no logarithmic corrections [41].

5.4 Transport in a single domain

2 In this section we consider transport in two spatial dimensions, with the isotropic dispersion εk = k /(2m)−

Λ/2. We will find that many aspects of the transport can be computed in a traditional Boltzmann transport computation, due to the large N and M limits. In particular, quantum corrections to transport, of the type leading to quantum interference and localization, are suppressed by the local disorder, the non-quasiparticle nature of the charge carriers, and the large number of fermion flavors.

In our double large N and M limit, if M/N = 0, the only vertex corrections to the uniform conductivities

† that aren’t trivially killed by this limit are the ones that involve uncrossed vertical ladders of fi fj propagators in the current-current correlator bubbles (First diagram of Fig. 5.2b). However, since the f propagators are purely local and independent of momentum, these diagrams vanish due to averaging of the vector velocity in

118 Figure 5.2: (a) The uniform current-current correlation bubble used to compute conductivities. The current vertices are black squares and the black lines are conduction electron (c) propagators. (b) and (c) Additional diagrams forming ladder series, with ladder units of up to 3 loops, that contribute to the conductivities and are not immediately suppressed by the large N and M limits. The red lines are island fermion (f) propagators that do not carry momentum. The dashed blue lines carry momentum and come from disorder averaging of the non-translationally invariant coupling x gijkl. These diagrams however vanish upon momentum integration in the loops containing the current vertices, for reasons mentioned in the main text. the current vertices over the closed fixed-energy contours in momentum space, as the scattering of the conduction electrons is isotropic, just like in the textbook problem of the non-interacting disordered metal [247].

Unlike the non-interacting disordered metal, there is no localization in two dimensions as the crossed-ladder

‘Cooperon’ diagrams are suppressed by the large M limit. Hence, the relaxation-time-like approximation of keeping only self-energy corrections is valid.

If M/N is nonzero but O(1) or smaller, then certain 3-loop and higher order ladder insertions (Such as

Fig. 5.2c) also contribute extensively in M to the current-current correlation. However, these diagrams again vanish due to the averaging of the vector velocity mentioned above. All this happens regardless of the values of g, J, Λ, µc, and for both energy and electrical currents.

5.4.1 Marginal-Fermi liquid

We first discuss a Boltzmann transport approach in the MFL regime. For simplicity, we consider infinite bandwidth and an infinitely deep Fermi sea. The uniform current-current correlation bubble (Fig. 5.2a) is

119 given by, for an isotropic Fermi surface,

∫ v2 ∑ ∞ ⟨ ⟩ − F dε 1 1 IxIx (iΩm) = M ν(0)T c c , 2 −∞ 2π iωn − ε − Σ (iωn) iωn + iΩm − ε − Σ (iωn + iΩm) ωn (5.28) where vF = kF /m is the Fermi velocity (on a lattice vF ∼ t, since the lattice constant a is set to 1). Using the spectral representation, this can be converted to give the DC conductivity

∫ ( ) v2 ν(0) ∞ dE E 1 σMFL = M F 1 sech2 1 . (5.29) 0 | c | 16T −∞ 2π 2T ImΣR(E1)

Inserting the self energy, we can scale out T and numerically evaluate the integral, giving

( ) v2 σMFL = 0.120251 × MT −1J × F cosh1/2(2πE). (5.30) 0 g2

MFL ≪ ≫ If we want σ0 /M 1, we must have T Tinc, implying a crossover into the IM regime. Thus the MFL is never a true bad metal, but its resistivity can still numerically exceed the quantum unit h/e2, depending on parameters.

MFL The ‘open-circuit’ thermal conductivity κ0 , which is defined under conditions where no electrical current flows, is given by MFL 2 MFL MFL − (α0 ) T κ0 =κ ¯0 MFL , (5.31) σ0

MFL MFL where κ¯0 is the ‘closed-circuit’ thermal conductivity in the presence of electrical current, and α0 is the thermoelectric conductivity. The thermoelectric conductivity vanishes when the temperature is much smaller than the bandwidth and Fermi energy, due to effective particle-hole symmetry about the Fermi surface, so

120 MFL MFL κ0 =κ ¯0 . The Lorenz ratio is then given by

∫ ∞ ( ) MFL MFL dE1 E2sech2 E1 1 MFL κ0 κ¯0 −∞ 2π 1 2 |Im[E1ψ(−iE1/(2π))+iπ]| L = = = ∫ ∞ ( ) = 0.713063 × L0, (5.32) σMFLT σMFLT dE1 sech2 E1 1 0 0 −∞ 2π 2 |Im[E1ψ(−iE1/(2π))+iπ]|

2 which is smaller than L0 = π /3 for a Fermi liquid.

In the presence of a uniform transverse magnetic field, we can use the following improved relaxation- time linearized Boltzmann equation (which incorporates an off-shell distribution function) for a temporally slowly-varying and spatially uniform applied electric field [105, 248], since there are no Cooperons in the large-M limit, and hence none of the typical localization-related corrections [249] to the conductivity tensor.

The Boltzmann equation reads (here, t is time, not the hopping amplitude, and B is a dimensionless version of the magnetic field B which shall be explained below)

− c ˆ· ′ ˆ×B ·∇ c (1 ∂ωRe[ΣR(ω)])∂tδn(t, k, ω)+vF k E(t) nf (ω)+vF (k zˆ) kδn(t, k, ω) = 2δn(t, k, ω)Im[ΣR(ω)],

(5.33) ( ) ω/T where nf (ω) = 1/ e + 1 is the Fermi distribution, δn is the change in the distribution due to the applied electric field, the conduction electrons are negatively charged, and the magnetic field points out of the plane of the system. This equation is derived in Appendix E.2 from the Dyson equation on the Keldysh contour, and can be solved by the ansatz δn(t, k, ω) = k · φ(t, ω) = kiφi(t, ω).

R In the DC limit, the effective mass enhancement (1 − ∂ωRe[Σ (ω)]) does not matter [248] (the effective mass enhancement is important for AC magnetotransport and affects the frequency at which the cyclotron resonance occurs; it shifts the cyclotron resonance from the cyclotron frequency defined by the bare mass to the one defined by the effective mass. The enhanced effective mass also appears in the specific heat [243] and Lifshitz-Kosevich formula [250] of MFLs). We then have

ˆ · ′ ˆ × B · ∇ c vF k E nf (ω) + vF (k zˆ) kδn(k, ω) = 2δn(k, ω)Im[ΣR(ω)], (5.34)

121 We note that in (5.34), B is dimensionless in our choice of units. Since the quantities we set to 1 were the magnitude of the electron charge e, the lattice constant a, and ℏ and kB, we have

2 B eBa = ℏ , (5.35)

i.e. the flux per unit cell in units of ℏ/e.

Substituting δn(k, ω) = kiφi(ω) into (5.34), we obtain

( ) −1 vF ′ c B vF φi(ω) = nf (ω) 2Im[ΣR(ω)]δij + ϵij Ej. (5.36) kF kF ij

Using the current density

∫ ∫ 2π ∞ dθ dω ˆ ˆ Ii = −Mν(0) vF kiδn(kF k, ω), (5.37) 0 2π −∞ 2π we get the longitudinal and Hall conductivities

∫ ( ) 2 ∞ − c MFL vF ν(0) dE1 2 E1 Im[ΣR(E1)] σL = M sech c 2 2 2 , 16T −∞ 2π 2T Im[Σ (E1)] + (vF /(2kF )) B ∫ ( ) R v2 ν(0) ∞ dE E (v /(2k ))B σMFL = −M F 1 sech2 1 F F . (5.38) H c 2 2B2 16T −∞ 2π 2T Im[ΣR(E1)] + (vF /(2kF ))

Note that, given the scaling of (5.13), these can be immediately written as

MFL ∼ −1 B MFL ∼ −B −2 B σL T sL((vF /kF )( /T )), σH T sH ((vF /kF )( /T )). (5.39)

The asymptotic forms of the functions sL and sH are

2 0 sL,H (x → ∞) ∝ 1/x , sL,H (x → 0) ∝ x . (5.40)

122 So we have obtained the advertised B/T scaling in the MFL regime. However, with the asymptotic forms noted above, it is not difficult to see that the magnetoresistance, ρxx saturates at large B. Nevertheless, the results above will be useful as inputs into our consideration of the effects of macroscopic disorder in

Section 5.5: we will show there that the B/T scaling survives, and the macroscopic disorder leads to a linear in B magnetoresistance.

We now show that the numerical scale of the B/T crossover is in general accord with the observations.

In (5.38), for the ‘Planckian’ choice of parameters described at the end of Sec. 5.3.2, B becomes ‘large’

c | | ≲ MFL (i.e., the cyclotron term in the denominators overwhelms Im[ΣR(E1)] for E1 T , causing σH to start

2 decreasing with increasing B), when eBa /ℏ ≳ kBT/t. Using reasonable values of the lattice constant a = 3.82 Å and the hopping t = 0.25 eV, the above inequality can also roughly be written as µBB ≳ kBT ,

2 where µB is the Bohr magneton, since a et/ℏ ≈ 0.96µB for these parameters.

In the analysis of the IM regime to follow, there is no such notion of ‘large’ magnetic fields; regardless of the value of B, the field-dependent corrections to the conductivity tensor remain much smaller than its zero-field value.

5.4.2 Incoherent metal

This subsection considers transport in the IM phase discussed earlier, in which the Fermi surface is washed out, and shows quantitatively that the orbital effects of a magnetic field on charge transport are strongly suppressed irrespective of the strength of the field. The physical reason for this effect is that the effective mean-free-path of the electrons in the IM is less than a lattice spacing, with conduction occurring locally and incoherently across individual lattice bonds. The effect of the Lorentz force on the electrons is thus negligible. If the reader is uninterested in the details of the following computations, they may move on to the next section.

123 In the IM regime we have

∫ ( ) 2 ∞ IM MΛ dE1 2 E1 c 2 σ0 = sech (A (k, E1)) . (5.41) 32πT −∞ 2π 2T

The spectral function is independent of k in the IM, and we decoupled the momentum integral implicit in the above equation, generating a prefactor of Λν(0)/(2π). For simplicity we consider M/N = 0 in this subsection. A small finite M/N only rescales Gc, as shown by (5.25, 5.22), and hence leads to no qualitative difference in any of the following results. We have

c 2π c 4π c + A (k, E1) ≡ A (E1) ≡ − Im[G (iωn → E1 + i0 )] Λν(0) Λν(0) ( ) [ − E ] E 1 − i(E1 2π cT ) − 3/4 1/4 2π c 1/2 1/4 E Γ − i( 1) π (i + e )J√ cosh (2π ) ( 4 2πT ) = 2Im E − E , (5.42) gT 1/2Λ1/2ν1/2(0) 1 + e4π c 3 − i(E1 2π cT ) Γ 4 2πT and we get ( ) 1/2 E IM 1/2 × −1 × Λ cosh (2π ) σ0 = (π /8) MT J 2 . (5.43) ν(0)g cosh(2πEc)

IM ≪ Due to the IM existing only at temperatures above Tinc, given by (5.24), we always have σ0 /M 1, which makes the IM a bad metal. Note that the slope of the resistivity ρ0(T ) = 1/σ0(T ) vs temperature in the IM generically differs from that in the MFL by an O(1) number, as can be seen by comparing (5.30) and (5.43).

IM IM IM The Lorenz ratio in the IM is (here, the thermoelectric conductivity α0 does not vanish, so κ0 and κ¯0 are distinct quantities)

∫ ( ) ∫ ( ) ∞ dE1 2 E1 c 2 2 ∞ [ −∞ E1sech (A (E1)) ] dE1 2 2 E1 c 2 − ∫ 2π ( 2 ) −∞ E1 sech (A (E1)) ∞ dE E 2π 2 1 sech2 1 (Ac(E ))2 IM −∞ 2π 2 1 3 L = ∫ ∞ ( ) = ×L0, regardless of E, Ec. dE1 2 E1 c 2 8 −∞ 2π sech 2 (A (E1)) (5.44)

This result was also obtained by a different method for the IM of Ref. [42], although they only analyzed the particle-hole symmetric case equivalent to Ec = 0.

124 Another dimensionless ratio that is interesting is the thermopower, i.e. the ratio of the thermoelectric to electrical conductivities,

∫ ∞ ( ) IM dE1 E sech2 E1 (Ac(E ))2 SIM α0 −∞∫ 2π 1 ( 2) 1 E 0 = = ∞ = 2π c. (5.45) σIM dE1 2 E1 c 2 0 −∞ 2π sech 2 (A (E1))

This relationship between the thermopower and the spectral asymmetry Ec was also found in a different model of coupled SYK islands realized in Ref. [41]. The ratios (5.44), (5.45) hold even for a finite small

M/N, as the effect of a finite small M/N is simply a rescaling of the Green’s function Gc.

Let us describe the fate of magnetotransport in the IM regime. On a lattice, we have Λν(0) ∼ 1. Then √ 2 2 2 JIM = g /J, and the conduction electron self-energy is ∼ JIMT . We have JIMT ≫ t ∼ Λ , so, to leading order we can neglect the dispersion in Fermion propagators. Then, there is nothing for the magnetic field to couple to, and consequently no magnetotransport.

To illustrate this, let us compute the correlator of currents in perpendicular directions in real space on a square lattice. The uniform current operators are

∑ ∑M 1 it † I (τ) ≡ I (τ) ≡ − c (τ)c (τ) + h.c., x V 1/2 rx 2V 1/2 r+ˆx,i ri r r; i=1 ∑ ∑M 1 it † I (τ) ≡ I (τ) ≡ − c (τ)c (τ)eiϕ(r) + h.c., (5.46) y V 1/2 ry 2V 1/2 r+ˆy,i ri r r; i=1

where we have used a gauge with the magnetic vector potential Ar pointing along the y direction, giving rise to the phase factors eiϕ(r) on bonds in the y direction. The system volume in units of the unit cell volume is

125 V . We then have

′ Tτ ⟨Ix(τ)Iy(τ )⟩

2 ∑ [ t † † ′ ′ ′ iϕ(r′) † † ′ ′ −iϕ(r′) = −M T ⟨c (τ)c (τ)c ′ (τ )c (τ )e ⟩ − T ⟨c (τ)c (τ)c ′ (τ )c ′ (τ )e ⟩ 4V τ r+ˆx r r +ˆy r τ r+ˆx r r r +ˆy rr′ ] † † ′ ′ ′ † † ′ ′ − ′ − T ⟨ ′ iϕ(r )⟩ T ⟨ ′ iϕ(r )⟩ τ cr(τ)cr+ˆx(τ)cr′+ˆy(τ )cr (τ )e + τ cr(τ)cr+ˆx(τ)cr′ (τ )cr +ˆy(τ )e , (5.47)

where we have dropped the sum over flavor indices in favor of a global factor of M, and T denotes time- ordering. To leading order in t, since the c Green’s functions are completely local,

† ′ ′ T ⟨ ⟩ ′ c − τ cr(τ)cr′ (τ ) = δrr G (τ τ ), (5.48)

none of the terms in (5.47) can be nonzero. Similarly, at O(t2), there is no field-dependent correction to the

⟨IxIx⟩ correlator.

Perturbing in t, in order for (5.47) to be nonzero, we need to insert hopping vertices in order to close the

4-point correlation functions of the c’s. To lowest order in t, this requires insertion of two hopping vertices into each of the 4-point correlation functions in (5.47), so that the connected contractions of c’s and c†’s into local c Green’s functions go around a single plaquette of the lattice. Again, due to our choice of gauge, hopping vertices along bonds in the y direction come with phase factors. But we obtain, as we should, a gauge-invariant answer for the connected part, which is of interest to us here (the electrons are negatively charged, and B is defined in terms of B as in Sec. 5.4.1)

∑ 4 c 3 c c ⟨IxIy⟩(iΩm) = −iM sin(B)t T [(G (iωn)) (G (iωn + iΩm) − G (iωn − iΩm))]. (5.49) ωn

At O(t4), vertex corrections from the coupling g to this leading contribution vanish due to the non-correlation

′ ≪ r r ≫ 2 ′ of g between distinct lattice sites, i.e. gijklgjilk = g δrr .

126 The DC Hall conductivity follows,

[ ] IM − 1 ⟨ ⟩ → + − ⟨ ⟩ → + σH = lim IxIy (iΩm ω + i0 ) IxIy (iΩm 0 + i0 ) ω→0 iω ∫ ∞ n (E ) − n (E ) B 4P dE1 dE2 c c f 2 f 1 = 2M sin( )t A3(E1)A (E2) 2 , (5.50) −∞ 2π 2π (E2 − E1) where P denotes the Cauchy principal value, and

 ( ) − E E 3 1 − i(E1 2π cT ) − 2π c 3 3/4 E Γ c c + 3 (i 1)(i + e ) cosh (2π ) 4 2πT  A (E1) ≡ −2Im[(G (iωn → E1+i0 )) ] = Im ( ) , 3 3/2 E i(E −2πE T ) 25/2π9/4J T 3/2(1 + e4π c )3/2 3 3 − 1 c IM Γ 4 2πT (5.51)

c 3 is the spectral function of (G (iωn)) . If Ec = 0, then the Hall conductivity vanishes due to the evenness

c c of the spectral functions A and A3. This corresponds to half-filling the square lattice, so this is expected.

Scaling out T and evaluating the integral numerically gives

4 E IM − B t cosh(2π ) IM E σH = M sin( ) 2 2 ΞH ( c), (5.52) JIMT

IM E E E E ∞ where ΞH ( c) is odd in c, positive for positive c, and vanishes when c = 0, . This is a very small contribution regardless of B; the already small flux per unit cell B is further multiplied by a small parameter

4 2 2 E O |E| t /(JIMT ). Note that we consider cosh(2π ) to be (1). If is very large, then the conduction electrons do not scatter effectively off the islands, as discussed before, and our perturbative expansion in hopping is no longer valid, and in that case the system is once again described by the MFL. For the Hall conductivity to

IM ∼ 2 B ∼ 2 ≫ be comparable to the longitudinal conductivity σ0 t /(JIMT ), we need sin( ) JIMT/t 1, which is not even mathematically possible.

Similarly, the field-dependent correction to the Ix-Ix correlator is

∑ 4 c 2 c 2 ∆B [⟨IxIx⟩(iΩm)] = −Mt cos(B)T (G (iωn)) (G (iωn + iΩm)) , (5.53) ωn

127 0.014 0.015 0.012 0.010 0.010 0.005 0.008 0.000 0.006 - 0.005 0.004 - 0.010 0.002 - 0.015 0.000 - 1.0 - 0.5 0.0 0.5 1.0 1.0- 0.5 0.0 0.5 1.0

IM E IM E Figure 5.3: Plots of (a) ΞH ( c) and (b) ΞL ( c). Both functions vanish in the limits of the fully filled and empty lattice (Ec = ∓∞ respectively), as they should. leading to the field-dependent correction to the longitudinal conductivity

∫ ( ) M t4 dE E ∆ [σIM] = cos(B) 1 Ac (E )sech2 1 , (5.54) B L 8 T 2π 2 1 2T where

 ( ) i(E −2πE T ) E 1/2 2 1 − 1 c (i + e2π c )2 cosh (2πE) Γ 4 2πT Ac (E ) ≡ −2 [(Gc(iω → E +i0+))2] = − i ( ) , 2 1 Im n 1 Im 3/2 4πE − E 2π JIMT (1 + e c ) 2 3 − i(E1 2π cT ) Γ 4 2πT (5.55)

c 2 is the spectral function of (G (iωn)) . Scaling out T and evaluating the integral numerically gives

4 E IM t cosh(2π ) B IM E ∆B[σL ] = M 2 2 cos( )ΞL ( c), (5.56) JIMT

IM E E E E → ∞ where ΞL ( c) is even in c, positive, nonzero for c = 0, and vanishes as c . The longitudinal conductivity is thus reduced when a field is applied, as is usually the case.

IM It is similarly thus not possible to get a field-dependent correction to σL that is comparable to its zero-field value. Thus we shall no more consider the IM regime for studying magnetotransport, as there is no qualitative difference between the regimes of ‘large’ and small B unlike in the MFL regime. For completeness, the plots

IM E of ΞH,L( c) are shown in Fig. 5.3.

128 Before we close this section, let us comment on the controllability of the hopping expansion used to compute the nonzero field-dependent conductivity corrections. Clearly, this hopping expansion must break down when t is large enough, as the MFL has a very different conductivity tensor. Going from (5.47) to

(5.49) and (5.53), we only kept those r′ relative to r that resulted in O(t4) corrections for the shortest closed paths from r to r′ and back. For arbitrary r′, one can draw infinitely many paths that go from r to r′ and back. These paths may also intersect themselves in general. For a path length l, there are < 4l paths for large l, as at each step, one has 4 choices of direction, and not all possibilities will result in a formation of the closed path from r to r′ and back. Each step involves mulitplying an additional local Green’s function

1/2 and factor of t, or roughly a factor of ∼ t/(JIMT ) ≪ 1 into the amplitude. Therefore, the total weight

1/2 l ′ of paths of length l should be < (4t/(JIMT ) ) . The total weight of all paths between r, r then is <

∑∞ (4t/(J T )1/2)l = (4t/(J T )1/2)lmin /(1 − 4t/(J T )1/2), where l is the length of the shortest l=lmin IM IM IM min

′ ′ 1/2 closed path between r, r , which scales as the lattice distance between r, r . Thus, for t/(JIMT ) ≪ 1, the expansion is well behaved: as r′ gets further away from r, the terms are exponentially suppressed in the distance between r and r′, whereas the number of r′’s a given distance away from r grows only linearly in that distance in two dimensions. Unsurprisingly, this is just the condition T ≫ Tinc we obtained earlier for the crossover into the IM regime.

5.5 Macroscopic transport via Effective-medium/Random-resistor

theory

We now return to the MFL with B/T scaling that was described in Section 5.4.1. We will show here that adding macroscopic disorder leads to a linear-in-B magnetoresistance at large B, while preserving the B/T scaling. We will treat the inhomogeneity in a classical transport framework. The quantum computation in

Section 5.4.1 is used to compute a local σxx and σxy, which is then in put into a computation of global

129 transport in a disordered sample by composing resistivities using Ohm’s and Kirchhoff’s laws.

5.5.1 Setup

We seek to understand the effects of additional macroscopic disorder on the transport of charge in the MFL at ‘large’ magnetic fields B, in two spatial dimensions. This additional macroscopic disorder leads to the variation of the local conductivity tensor σ(x) across the sample. Since the conduction electrons in our model interact with valence electrons in the islands through a non-translationally invariant interaction mi- croscopically, the Navier-Stokes equation of hydrodynamics that describes dynamics of a nearly-conserved macroscopic momentum [159] is not applicable to us, since this requires microscopic equilibriation of the electron fluid through momentum-conserving interactions (the effects of weak disorder on the magnetoresistance of a generic electron fluid with macroscopic momentum were studied in Ref. [251]; they did not find any regimes of linear magnetoresistance, instead finding that the magnetoresistance was quadratic with a prefactor controlled by the fluid viscosity). Thus, at the coarse-grained level, we just have the equation for charge conservation, and Ohm’s law

∇ · I(x) = 0, I(x) = σ(x) · E(x), E(x) = −∇Φ(x). (5.57)

The effective local electric field E(x) (which includes the effects of Coulomb potentials generated due to charge inhomogeneities [155]) fluctuates spatially due to the macroscopic disorder, but equals an applied ∫ ⟨ ⟩ ≡ 1 2 external electric field E0 = E(x) V d x E(x) on spatial average. We define the global conductivity

e e e tensor σ through the relation ⟨I(x)⟩ = σ · E0, and parameterize the deviation σ(x) − σ = δσ(x). The condition ⟨I(x) − ⟨I(x)⟩⟩ = 0 then gives ⟨χ(x) · E0 ≡ δσ(x) · E(x)⟩ = 0.

Following Ref. [237], without making any additional approximations, the solution of these equations can

130 be formally cast in the form

∫ 2 ′ ′ ′ ′ ′ ′ Φ(x) = −E0 · x + d x G(x, x )∇ · (δσ(x ) · ∇ Φ(x )), (5.58)

where the Green’s function satisfies ∇ · (σe · ∇G(x, x′)) = −δ(x − x′), G(x, x′) = G(x′, x), and G(x, x′ ∈

∂V ) = 0, for the system boundary ∂V , which we take to infinity. Taking a gradient on both sides, we get

∫ 2 ′ ′ ′ ′ ′ E(x) = E0 − d x [(δσ(x ) · E(x )) · ∇ ] · ∇G(x, x ), or ∫ χ(x) = δσ(x) − δσ(x) · d2x′ K(x, x′) · χ(x′), (5.59)

where the second line follows from the first by left-multiplying both sides by δσ(x), and then demanding

K ′ ′ G ′ that it hold for any E0, and ij(x, x ) = ∂i∂j (x, x ).

We now assume that the disorder divides the sample into macroscopic domains whose size is much smaller than the sample size, but much bigger than the smaller of the electron mean-free path and electron cyclotron radius, and the tensors χ and δσ take on constant values in a given domain. For a given domain p, we can write

∫ ∑ ∫ ′ χp = δσp − δσp · d2x′ K(x ∈ p, x′) · χp − δσp · · d2x′ K(x ∈ p, x′) · χp . (5.60) ′ p p′≠ p p

For the second integral over domains other than the given domain, we replace χn with its spatial average

⟨χ⟩. This is the ‘effective-medium’ approximation [237]: The equivalent conductivity of each domain is controlled in part by a ‘mean-field’ of domains surrounding it. However, since our conventions are set up so that ⟨χ⟩ = 0, this second term drops out. Then, spatially averaging both sides, we obtain

∑ ∑ V pχp = 0 ⇒ V p(I + δσp ·Mp)−1 · δσp = 0, (5.61) p p

131 H p Mp G ′ ′p where V is the volume fraction of domain p and ij = ∂′p ∂i (x, x )ˆnj , where the integral is over the primed coordinate, and nˆ′p is the outward-pointing unit normal vector on the boundary of p, varying with the primed coordinate.

If the local conductivity tensor σ(x) is known in all domains, (5.61) can then be solved for σe. In our

e e − e e e two-dimensional electron problem, we expect σij = δijσL ϵijσH , where σL is even in B and σH is odd in

G ′ − | − ′| + e B because of Onsager reciprocity, so we obtain the Green’s function (x, x ) = ln( x x 0 )/(2πσL).

Mp e Then, for circular domains, ij = δij/(2σL) is indeed independent of x. This makes (5.60) and (5.61) self-consistent [237]. For other domain shapes, there are corrections when x is near the domain boundary.

For an analytically solvable toy model, we assume that the σ(x) can take either of two possible values

σa and σb in circular domains that are spatially randomly distributed over the sample [236, 240] (Fig 5.4a).

As far as the asymptotic low and high-field magnetoresistance goes, this already yields the same qualitative behavior at large and small fields as a more complicated model with a distribution of different types of domains [242]. Furthermore, the ‘mean-field’ like effective-medium approximation has also been shown to produce results for the magnetoresistance equivalent to exact numerical solutions of (5.57) in random- resistor network models [238, 239, 242]. In the simplified two-type scenario (5.61) then simplifies to [240]

( ) ( ) a − e −1 b − e −1 a I σ σ · a − e − a I σ σ · b − e V + e (σ σ ) + (1 V ) + e (σ σ ) = 0. (5.62) 2σL 2σL

If V a = 1/2, this yields an unsaturating high-field linear magnetoresistance [240]. For the model with a distribution of domains, the equivalent condition is that the distribution is symmetric about its mean [242].

For V a detuned from 1/2, the magnetoresistance saturates, but there is an intermediate regime of fields in which the magnetoresistance is approximately linear, and the saturation field becomes arbitrarily large as V a approaches 1/2 [240]. The rough reasoning behind the saturation appears to be that, if one type of domain is far more common than the other, the current flowing through the sample mainly finds paths involving only

132 one type of domain, and hence the global magnetoresistance behaves like that of a single domain, which saturates at high fields [239]. We will do our analysis with the symmetric distribution V a = 1 − V a = 1/2.

A physical picture for the high-field linear magnetoresistance was provided in Ref. [238], and involves the contribution of the local Hall resistance (which is linear in B) to the global longitudinal resistance due to the distortion in current paths arising from spatial fluctuations of the local Hall resistance: In a uniform sample, charge accumulation at the edges of the sample parallel to the applied electric field produces a global Hall electric field perpendicular to the applied electric field that cancels out Hall currents throughout the sample.

On the other hand, if the sample has a disordered local conductivity tensor, the global Hall electric field no longer cancels out local Hall currents throughout the sample. Thus, the global longitudinal resistance becomes dependent on the local Hall resistances.

5.5.2 Application

c We note that in (5.38), the sech is strongly peaked near E1 = 0, whereas for a finite temperature, Im[ΣR(E1)] does not vary drastically with E1 near E1 = 0 over the range which the sech is appreciable. We can thus

c replace Im[ΣR(E1)] with γ/2 from (5.15). Regardless of this approximation, we note from (5.38) that

MFL ∼ 2 MFL ∼ σL T/B and σH 1/B at large B, which is what the effective-medium theory needs to produce linear magnetoresistance at large B. This asymptotic scaling holds even if we had multiple MFL bands, thus adding their conductivity tensors to get the appropriate local conductivity tensor.

We thus input the following conductivity tensors into the effective-medium calculation (we take the band mass m = kF /vF to be the same in both types of electron-like domains a and b):

( ) MFL a,b σ0a,b B σij = 2 2 δij + ϵij . (5.63) 1 + B /(mγa,b) mγa,b

The scattering rate γ can fluctuate across domains due to fluctuations in g, induced by fluctuations in the

133 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.000 0.005 0.010 0.015

Figure 5.4: (a) A cartoon of a two-dimensional sample with a random distribution of approximately equal fractions of two types of domains, for which an exact analytic solution of the effective-medium equations for magnetotransport is possible. The magnetic field B points out of the plane of the sample. (b) Plots of the normalized change in global longitudinal resistance due to dimensionless magnetic field B (orange) and due to temperature T (blue), obtained from F F B 2 (5.65). We use Eb /Ea = 0.8 and γb/γa = 0.8. The dimensionless magnetic field is the flux per unit cell Ba in ℏ ∼ F B units of /e (5.35). We use m = 0.005 1/Ea,b. The orange ( ) curve is evaluated at T = 1.0 and γa = 0.1 and the blue (T ) curve is evaluated at B = 0.0025 and γa = 0.1T . The curves are slightly offset for visualization, but actually lie on top of each other, demonstrating a scaling between magnetic field and temperature. Both the B and T dependencies are quadratic at small fields or temperatures and cross over to linear at large fields or temperatures.

MFL densities of islands, and the base conductivity σ0a,b can fluctuate across domains due to fluctuations in both g and in the electron density. Then, solving (5.62) for V a = 1 − V a = 1/2, we get the global longitudinal and Hall resistances respectively,

√ ( ) ( ) 2 2 e B 2 MFL − MFL 2 2 MFL MFL σ ( /m) γaσ0a γbσ + γaγ σ0a + σ ρe ≡ L = 0b ( b ) 0b , L e2 e2 MFL MFL 1/2 MFL MFL σL + σH γaγb(σ0a σ0b ) σ0a + σ0b σe /B e ≡ − H (γa + γb ) ρH e2 e2 = MFL MFL . (5.64) σL + σH mγaγb σ0a + σ0b

e B − e The magnetoresistance ρL( ) ρL(0) is thus linear as promised at high fields, and is quadratic at low fields.

2 Considering the isotropic parabolic dispersion εk = k /(2m)−Λ/2, and using (5.30), (5.15), and ν(0) =

MFL a,b a,b 2 m, we can write σ0a,b = MwσEF /γa,b, where wσ = 0.135689 and EF = mvF a,b/2 are the Fermi

134 energies. We can then rewrite (5.64) as

( ) ( ) 2 1/2 B 2 (1−EF /EF ) γ2 + b a a m F F 2 (γb/γa+E /E ) (1 + γ /γ ) w ρe = b a , w ρe = b a . (5.65) σ L 1/2 F F 1/2 σ H F F F M(γa/γb) (Ea Eb ) MmEa (Eb /Ea + γb/γa)

e B Plots of the normalized change in ρL due to and T are shown in Fig. 5.4b. This simplified model with two types of domains thus leads to a global longitudinal resistance that adds T and B in quadrature*, as seen in the experiment of Ref. [17]. A continuous gaussian distribution of electron densities across the domains will also yield a qualitatively similar scaling function to the above quadrature function [242]. In general, the zero- field linear-in-T and high-field linear-in-B behavior (as well as the scaling between B and T ) will emerge universally from such resistor-network models, but the interpolation between the two regimes is sensitive to the distribution of the local conductivity tensors.

e The Hall resistance is ρH is sensitive to the disorder distribution and thus is not trivially controlled by

∝ F F the average carrier density Mm(Eb + Ea )/2 even for the isotropic Fermi surfaces we consider, unless

e γa = γb. In this simplified version of the problem, ρH is independent of temperature. However, we expect

e that more complicated disorder distributions generically give rise to some temperature dependence of ρH , which would depend on the disorder distribution even at a qualitative level. A detailed analysis of such effects is beyond the scope of the present work, and will be considered in the future.

Since γa,b ∝ T , the crossover from quadratic to linear magnetoresistance occurs at a field scale proportional to temperature. Additionally, if we use the ‘Planckian’ choice of parameters, and if the disorder

| − F F | F F O distribution is such that 1 Eb /Ea /(γb/γa + Eb /Ea ) is an (1) number, the crossover occurs at a field scale given by µBB ∼ kBT , as discussed at the end of Sec. 5.4.1. While this is most definitely a fine-tuned situation, and would require substantial variation in the charge densities between domains, it is within the

F F | − F F | O scope of our theory. Alternatively, if γa(γb/γa + Eb /Ea )/(kBT 1 Eb /Ea ) is an (1) quantity (but *Holographic realizations of a variety of magnetoresistance scalings, including quadrature, were found in Ref. [252].

135 ∝ e γa T is much smaller than kBT ), then ρL can still be controlled by the approximate scaling function √ 2 2 1 + (µBB) /(kBT ) for much smaller variations in the charge densities between domains.

The effective-medium theory is applicable when the domain sizes are much greater than the smaller of the electron mean free path and electron cyclotron radius in a single domain. At low temperatures and weak fields, electrons can move through a domain without significant loss or deflection of momentum, and the effects of scattering off the boundaries between domains then become important, adding a temperature- independent residual resistivity to the result of the above computation.

In our analysis, we have neglected the effects of the feedback of heat currents on charge transport. In general, one would have an additional analogous set of equations to (5.57) for heat currents and temperature gradients in place of charge currents and electric fields. Since there is no concept of bulk fluid motion due to translational symmetry breaking at the microscopic level, the equations for heat currents and charge currents would only be coupled if the local thermoelectric tensor α(x) were nonzero. However, in the MFL, with

≪ F T Ea,b, α(x) is negligible as discussed in Sec. 5.4.1, and our decoupled analysis of charge currents is hence still applicable. Somewhere in the crossover region between the MFL and the IM, a regime may exist where both α(x) and the effects of magnetic fields on the local conductivity tensors are simultaneously significant, and there may be a significant feedback of thermoelectric effects on the charge magnetotransport.

We leave a detailed study of such effects for future work.

5.6 Discussion

The strange metal phases of the cuprate and pnictide high-Tc superconductors occur at finite dopings, and consequently display significant amounts of disorder. Experimentally, there is direct evidence for disorder at (i) microscopic levels, due to irregular placements of dopant atoms [253], and (ii) meso- and macroscopic levels, due to a variety of factors ranging from crystalline imperfections to charge puddles caused by impuri-

136 ties and non-isovalent dopants [254, 255]. Additionally, due to these materials being layered, with relatively poor interlayer conductivities, imperfections in a layer may further induce heterogeneities in the charge distributions of adjacent layers through Coulomb forces.

We have attempted to paint an impressionist picture of transport and magnetotransport in a strange metal by developing a solvable model that incorporates disorder at both microscopic and macroscopic levels. At the microscopic level, we built off remarkable recent developments [40–42, 235, 256, 257] in realizing solvable field-theoretic descriptions of extended non-Fermi liquid phases using SYK models. These models couple together SYK quantum islands without quasiparticle excitations, and show how this can lead to non-Fermi liquid transport in an extended finite-dimensional phase. In our model we locally and randomly couple mobile conduction electrons to immobile quantum islands described by SYK models in a particular way. In this manner we realized a disordered marginal Fermi liquid (MFL) phase at low temperatures with a linear-in-T resistivity, and an identifiable Fermi surface. We determined the two-point functions, conductivities, and magnetotransport properties of this phase exactly in two spatial dimensions, finding a scaling between magnetic field and temperature in the conductivity tensor. Additionally, we showed that nearly- local ‘incoherent-metal’ (IM) phases, with no identifiable Fermi surface, are also realized in our model at higher temperatures in certain parameter regimes; these IMs can also have linear-in-T resistivities, but have very weak effects of magnetic fields on their charge transport properties, making them unlikely candidates for a description of the strange metals seen in experiments at lower temperatures, which is where the large linear-in-B magnetoresistances are also observed. However, the IMs may still be the correct concept at high temperatures, due to strong bad-metallic behavior displayed through their large resistivities, as is seen in experiments. It should also be noted that the large linear magnetoresistances are not observed in experiments performed at high temperatures where the system is a bad metal, with a zero-field resistivity much larger than the quantum unit h/e2 [17, 19], which is consistent with the behavior of an IM.

While the MFL regime of our model does indeed have a linear-in-T resistivity, and also a B/T scaling at

137 approximately the observed B scale, it yields a magnetoresistance which saturates at large B. To obtain a non-saturating magnetoresistance, we argued for the importance of macroscopic disorder in the MFL regime.

To model such effects, we applied the effective-medium approximation to a sample containing domains of our disordered linear-in-T MFLs with varying electron densities. While the effective-medium approximation is a mean-field theory at the level of Kirchhoff’s and Ohm’s laws for current flow, it has shown to be equivalent to exact numerical simulations of random-resistor networks for magnetotransport [242], and has also had remarkable successes in describing experimentally observed magnetoresistances in other two-dimensional disordered materials [242, 258, 259]. For certain simplified disorder distributions, the effective-medium equations for magnetotransport are analytically solvable. These exactly solvable equations yield, in our case, a magnetoresistance that is quadratic in field at low fields, crosses over to linear in field at high fields, and is controlled by a scaling function between field and temperature, as seen in recent experiments on the pnictide and cuprate strange metals [17, 19].

On the experimental front, the anomalous high-field linear magnetoresistance in the cuprate and pnictide strange metals is already known to be dependent on the component of the magnetic field perpendicular to the sample plane [260], a feature that our model reproduces, since it is based on orbital effects of the magnetic field on charge transport. Furthermore, a strong linear component of the high-field magnetoresistance is seen even away from the critical doping at which the zero-field resistance is almost exactly linear-in-T [17, 19].

The disorder based mechanism considered by us would be consistent with this observation, as the zero-field linear-in-T behavior is not a prerequisite for high-field disorder-induced linear magnetoresistance; all that is required is that the local conductivity tensor behaves like (5.63) as a function of magnetic field.

On the theoretical front, we have been able to analytically calculate non-trivial magnetotransport properties in a somewhat contrived, but solvable, model of a disordered non-Fermi liquid. Studies along the lines of

Refs. [231–233] could show how such models emerge naturally as effective theories of realistic, disordered, single-band Hubbard models. We hope that our study motivates further investigations into the interplay of

138 disorder and strong interactions in the transport properties of the strange metal phases of the pnictides and cuprates.

139 Nothing is so painful to the human mind as a great and sudden change.

Mary Wollstonecraft Shelley

6 Coherent superconductivity with a large gap

ratio from incoherent metals

6.1 Introduction

Superconductivity in correlated systems often emerges from a mysterious incoherent metallic (IM) state with T -linear resistivity. The origin of the T -linear resistivity has been a subject of active research and debate [16, 228, 261, 262]. Moreover Refs. [216, 263, 264] have pointed out that superconductivity emerging out of such strange metals should be qualitatively different from that emerging out of conventional metals.

Nevertheless the lack of a solvable microscopic model has prevented the community from forming a concrete

140 connection between many inexplicable properties of the superconducting state and the IM state in correlated systems.

Recent proposals of microscopic models exhibiting IM transport in a solvable limit [40–42, 256, 265–

267], present new avenues. The approach shared among these models is to build on [35] finding non-Fermi liquid Green’s functions in a solvable model of fermions with infinite-range interactions. Although both this original model and a simpler model with Majorana fermions [36] exhibit non-Fermi liquid properties as well as interesting connections to quantum gravity [36, 37] in the solvable limit, they do not support local current operators. However, by introducing local coupling between multiple copies of these infinite ranged models, in the spirit of weakly coupled quantum dots each hosting multiple orbitals, Refs. [40–42, 256, 265, 266] established solvable microscopic models with IM transport. These models led to new insights regarding loss of quasiparticle coherence during scattering leading to such transport. But moreover, they have put us in an opportune moment to theoretically study the properties of superconducting (SC) phases born out of such

IMs, in a solvable limit.

In this chapter, we consider two models that can be solved in a large-N limit that demonstrate the much sought after transition from an IM with T -linear resistivity to SC. We then study the implication of strong correlations destroying coherent quasiparticles on the superconducting transition and state. In spite of the incoherent normal state, the paired state still supports a coherent supercurrent. We further show that a key prediction of the Bardeen-Cooper-Schrieffer (BCS) mean-field theory [4] of superconductivity is violated: the ratio between the zero temperature gap ∆ and the transition temperature Tsc far exceeds the BCS value of 2∆/Tsc ≈ 3.53. We compare this mechanism of gap ratio enhancement with that in the Eliahsberg theory and in experiments.

141 6.2 Model 1

We consider a lattice model of two species of fermions a, b with disordered local on-site Sachdev-Ye-Kitaev interactions of 4th order (SYK4), but with a uniform quadratic hopping, and an attractive term that pairs the two species locally (Fig. 6.1). It is given by

∑ ∑N ( ) ∑ ∑N ( ) † † † H = Kam a a a a + (a ↔ b) − t a a + (a ↔ b) + H.c. 1 i1,..,i4 i1m i2m i3m i4m im in m i1,..,i4=1 ⟨mn⟩ i=1 ∑ ∑N U † † − b a a b , (6.1) N im im jm jm m i,j=1 where m and n are the site indices with N fermions of each type. Here the disordered complex Gaussian random couplings Kαm satisfy ≪ Kαm Kβn ≫= K2/(8N 3)δ δ , where ≪ .. ≫ denotes i1,..,i4 i1,i2,i3,i4 i4,i3,i2,i1 αβ mn disorder-averaging, and all other averages are zero. A simpler model, without the attractive U term, and with only one type of fermion, was first proposed in Ref. [256].

Without the U term, and with K ≫ t, this model exhibits a crossover between a high temperature IM state with T -linear resistivity to a low tempterature Fermi liquid state. Specifically, for K ≫ T ≫ t2/K, the Green function is asymptotically given by the local SYK Green’s function in imaginary time [37]:

( ) π1/4 T 1/2 Ga,b(0 < τ < β) = − √ . (6.2) K1/2 2 sin(πT τ)

Using the Kubo formula, one can derive the linear-in-T resistivity in the large-N limit from the scaling of the above Green’s function [42, 265, 266]. On the other hand, for T ≪ t2/K, the Green function approaches that of a Fermi liquid whose resistivity scales as T 2 [256, 266].

The attractive U term leads to a spatially uniform s-wave pairing instability at T = Tsc. Once SC is ∑ ⟨ ⟩ established, the order parameter ∆0 = i aimbim /N condenses. In the large-N limit, we then get the

142 Dyson and gap equations (Appendix F)

∫ d G d k G(iωn, k) − ′ − 2G2 − ′ G ′ − (iωn) = d 2 2 2 , Σ(τ τ ) = K (τ τ ) (τ τ), (2π) 1 + U |∆0| |G(iωn, k)| ∫ ∑ d | |2 −1 − − d k G(iωn, k) ∆0 ∆0 G (iωn) = iωn ξk Σ(iωn),T d 2 2 2 = , (6.3) (2π) 1 + U |∆0| |G(iωn, k)| U ωn

−1 which can be iterated numerically starting with an infinitesimal ∆0 and the free fermion G(iωn) = (iωn) ∑ ∑ G T ⟨ † ⟩ T ⟨ † ⟩ in order to determine both G and ∆0. Here (τ) = τ i aim(τ)aim(0) /N = τ i bim(τ)bim(0) /N is the local time-ordered Green’s function, and ξk is the dispersion of the fermions.

2 For simplicity, while still capturing the essential physics, we consider d = 2, and ξk = Λk /(4π) −

Λ/2 ≡ ϵk − Λ/2, with ϵk ∈ [0, Λ], where Λ ∼ t is the bandwidth of the dispersion. We can then replace ∫ ∫ d2k → 1 Λ/2 (2π)2 Λ −Λ/2 dξk, and perform all momentum integrals analytically. In general, we determine Tsc numerically by taking the limit ∆0 → 0 in the last line of (6.3). For T > Tsc, the solution of (6.3) with

∆0 = 0 corresponds to a stable local minimum of the free energy. The large-N limit strongly suppresses fluctuations of ∆0 out of this minimum. As T is lowered below Tsc, the curvature of the free energy as a function of ∆0 at ∆0 = 0 changes sign, and the system condenses to a new minimum with ∆0 ≠ 0

(Appendix F).

2 When U is infinitesimal so that Tsc ≪ t /K, the SC arises out of a Fermi liquid, and we find the standard

2 BCS result of 2∆ = 3.53Tsc. On the other hand if both K and U are large such that K ≫ Tsc ≫ t /K, we obtain the transition to SC from the linear-in-T IM. From (6.3) and (6.2) we get (Appendix F)

( ) 2K 1/2 T ≈ tan−1 e−π K/U , (6.4) sc π where we employed a UV frequency cutoff ∼ K in (6.3). By solving (6.3) numerically, we can study how the gap ratio evolves through the crossover between SC emerging from a Fermi liquid to SC emerging from a linear-in-T IM. For this, we study the variation of the zero-temperature gap to single-particle excitations,

143 16

4 BCS Limit 0 0.001 0.010 0.100 1 10

2 Figure 6.1: Left: A cartoon of Model 1. Right: A plot of 2∆/Tsc as T → 0, vs KTsc/Λ for K = 1000, Tsc = 10, for different values of the bandwidth Λ ∼ t in Model 1. The value of U is adjusted as Λ is varied in order to keep Tsc fixed. For large Λ, the transition to SC is from a dispersive Fermi liquid, and we find the BCS result 2∆ ≈ 3.53Tsc. 2 For small values of Λ, such that KTsc/Λ ≫ 1, the transition to SC is from a non-Fermi liquid IM with a linear-in-T resistivity, and 2∆ ≫ 3.53Tsc.

∆, with the bandwidth Λ, keeping K and Tsc fixed. ∆ corresponds to the location of the peak of the spectral function A(ω, {k : ξk = 0}), with

[ ] G (ω, k) A(ω, k) = −2Im R , (6.5) 2| |2 ∗ − 1 + U ∆0 GR(ω, k)GR( ω, k) which may be obtained from a numerical solution of the real-time version of (6.3) (Appendix F). For small interactions and Tsc (relative to the bandwidth), SC emerges from a Fermi liquid, and the gap ratio is consistent with the “BCS” value of 2∆ ≈ 3.53Tsc. However, the interactions U and K are cranked up relative to the bandwidth so that SC emerges directly from a T -linear IM, the gap ratio substantially exceeds the BCS gap ratio (Fig. 6.1). When the gap ratio is enhanced significantly, we also find that the superconducting transition becomes first order, i.e., the order parameter ∆0 jumps discontinuously to a nonzero value at T = Tsc.

First-order transitions have also been noted earlier in studies of superconductivity arising from non-Fermi liquids [268, 269].

144 6.3 Model 2

We now consider a model that realizes an instability to SC from a non-Fermi liquid even for infinitesimal values of U. In order to avoid a Fermi liquid state as T → 0, we need a model that has no quadratic terms in its Hamiltonian. We however still want a linear-in-T resistivity above some small temperature scale. We thus replace the on-site interactions of Model 1 with higher order SYK8 terms, and the quadratic hopping between adjacent sites by pair hoppings that realize SYK4 interactions between adjacent sites. As we shall explain in detail, the scaling dimensions of the current operator and the local Green’s functions then lead to a linear-in-

T resistivity above a certain temperature. Since the charge transfer between sites is now strongly disordered, we have to use an attractive interaction given by not a conventional on-site pairing term, but rather a spatially uniform term that simultaneously binds a − b pairs on site and hops them between nearest-neighbor sites, which allows coherent pair hopping below Tsc and hence establishes superfluid phase coherence in the SC state.

We start with a single-site SYK8 model with two species of fermions:

( ) ∑N † † † † H = J a a ..a a ..a + J b b ..b b ..b , (6.6) 2,0 i1,..,i8 i1 i4 i5 i8 i1,..,i8 i1 i4 i5 i8 i1,..,i8=1

J α ≪ J α J α∗ ≫= J 2/(2304N 7) with complex Gaussian random couplings i1,..,i8 satisfying i1,..i8 i1,..i8 , with all other averages being zero. For J ≫ T , the resulting SYK8 Green’s function is [41]

( ) C T 1/4 Ga,b(0 < τ < β) = − 8 , J 1/4 sin(πT τ) ( ) ( ) ( ) π 1 7 C = sin1/4 Γ1/8 Γ1/8 . (6.7) 8 8 4 4

We then place a system described by (6.6) on each site of a lattice indexed by m. The random SYK8

145 Figure 6.2: (a) A schematic representation of Model 2. (b) Crossover diagrams in different regimes. For J ≫ K ≫ U, there is first a crossover to an IM with T -independent resistivity, before the SC transition. For J ≫ U ≫ K, we just have a transition from an IM with T -linear resistivity to an SC. couplings are not correlated between sites. We introduce two inter-site terms between nearest-neighbor sites: a random SYK4 interaction that hops a and b fermions independently in pairs between nearest-neighbor sites which is necessary for IM transport, and a uniform hopping term for a − b Cooper pairs which drives superfluid phase coherence below Tsc (Fig. 6.2(a)):

∑ ∑N ( ) † † H = J am a ..a a ..a + (a ↔ b) 2 i1,..,i8 i1m i4m i5m i8m m i1,..,i8=1 ∑ ∑N ( ) ∑ ∑N ( ) a⟨mn⟩ † † U † † + K a a ai nai n + (a ↔ b) + H.c. − b a ajnbjn + H.c. , i1,..,i4 i1m i2m 3 4 zN im im ⟨mn⟩ i1,..,i4=1 ⟨m,n⟩ i,j=1

(6.8)

≪ α⟨m,n⟩ α⟨m,n⟩∗ ≫ 2 3 with Ki1,i2,i3,i4,Ki1,i2,i3,i4 = K /(8zN ), and all other averages are zero. Note that the role of the

SYK4 inter-site interactions in this model is distinct from the intra-site SYK4 interactions in Model 1. The coordination number of the regular lattice is z.

Normal state of Model 2 – For T > Tsc, due to the large-N limit, the Dyson equation for the fermion

Green’s functions is local in space and is simply given by

Σ(τ) = −J 2G4(τ)G3(−τ) − K2G2(τ)G(−τ),

−1 G (iωn) = iωn − Σ(iωn), (6.9)

146 for both fermion types a and b. Defining the energy scaling dimensions [a] = [b] = 1/4 using the K terms in (6.8), we see that J is irrelevant at low energies, but dominates at high energies. This implies a crossover

2 between two distinct IM states around T ≈ K /J: An SYK8 dominant regime with the Green’s function

2 given in Eq. (6.7) for T ≫ K /J and an SYK4 dominant regime with the Green’s function given in Eq. (6.2) for T ≪ K2/J. Depending on the strength of the attractive interaction U superconductivity will set in out of IM states with qualitatively different transport.

The current operator on the bond indexed by ⟨mn⟩, that leads to a conductivity not suppressed by 1/N is

∑N mnˆ a⟨mn⟩ † † ↔ − Im = 2i (Ki1,..i4 ai1mai2mai3nai4n + (a b) H.c.), (6.10) i1,..,i4=1

(there is another contribution to the current from the U term, but it leads to a contribution to the conductivity that is not extensive in N in the IM). Using this, we then obtain the uniform, disorder averaged, current- current correlator at large-N

2 ˆ ˆ NK ≪ ⟨IlIl⟩(q = 0, τ) ≫= 4 G2(τ)G2(−τ). (6.11) z

2 For T ≫ K /J, and we can approximate G to be the SYK8 Green’s function (6.7) to obtain

NK2 σJ = 2C4 . (6.12) DC 8 zJT

Hence this regime is an IM with T -linear resistivity. However for T ≪ K2/J, the system will cross over to

K transport controlled by the SYK4 Green’s function (6.2) with σDC = N/z (i.e., a T -independent constant).

Depending on the relative strength of the attractive interaction U in comparison to the SYK4 interaction strength K, this T -independent resistivity IM may or may not be visible (see Fig. 6.3).

Superconductivity of Model 2 – The attractive U term leads to a leading uniform (q = 0) s-wave pairing

147 6 0.5

5 0.4 4 0.3 3 0.2 2 1 0.1

0 1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5

+ Figure 6.3: Left panel: Temperature dependence of the DC resistivity (ρ(T )/ρ(Tsc + 0 )), vs T/Tsc in Model 2. (i) J ≫ K > U, with a transition from a roughly T -independent resistivity to an SC (black), (ii) J ≫ U ≫ K, showing a direct transition from T -linear resistivity to an SC (red). The values of the parameters used are (i) J = 100, K = 20, U = 13.8584 and (ii) J = 100, U = 9.0515, K = 1. Tsc = 1 in both cases. Right panel: A plot of the normalized physical gap ∆/U as T → 0 in Model 2, obtained from the spectral function A(ω) determined numerically (Appendix F), vs the lower bound ∆b = 2Tsc/U (dashed line) for J = 100, K = 1, and different values of U. instability in the IM phase, which can be seen by considering the renormalization of the U term of (6.8) in the pairing channel at different values of the external momentum q, through the standard resummation of pairing bubbles. For infinitesimal U with J ≫ K ≫ U, we have a transition from an IM with an approximately

T -independent resistivity to an SC. In this case the physics of SYK4 controls Tsc and Tsc takes the same form as that in Model 1, given by Eq. (6.4). A new case of interest is accessible when J ≫ U ≫ K. In this regime the superconducting transition occurs in the temperature range with T -linear resistivity and we obtain (Appendix F) C4Γ2(1/4) U 2 T = 8 . (6.13) sc πΓ2(3/4) J

At Tsc, we then have a transition from an IM with a linear-in-T resistivity to an s-wave SC.

Now we can investigate implications of the IM normal state on the superconducting state. We find the correlation driven IM normal state affects the superconducting state through an enhancement of the gap ratio as in Model 1. To see this without a BCS limit to benchmark, we consider the limit of vanishing SYK interactions, i.e., U ≫ J, K. In this limit, the paired state becomes entropically unstable above transition temperature of Tsc = U/4, and the normal state contains featureless free fermions. Further one can find analytically that the zero temperature gap in this limit to be given by ∆b = U/2 = 2Tsc. Now the implication of IM normal state is apparent in the numerically obtained value of ∆ (see Fig. 6.3). ∆ always exceeds the

148 lower bound value of ∆b = 2Tsc (dashed line) in the presence of the J, K interactions. As in Model 1, we find that the superconducting transition becomes first order when the J, K interactions are strong enough to enhance the gap ratio significantly.

Our models also show coherent superfluid transport despite incoherence driven by the SYK interactions in the normal state. In the SC phase of Model 2, charge transport for T ≪ ∆ is controlled by gapless low

iθm energy phase fluctuations, whose Hamiltionian is derived by letting ∆0m = ∆0e ,

∫ NU ∑ NU H ≈ |∆ |2 (θ − θ )2 → |∆ |2 (∇θ)2. (6.14) θ z 0 m n z 0 ⟨m,n⟩ x

This implies the usual diamagnetic electromagnetic response at frequencies |ω| ≪ ∆ and T = 0 [270], with

2 NU |∆0| lim σij(ω, q = 0) ≈ 4δij , (6.15) ω→0 z iω and a superfluid phase stiffness that is extensive in N, uninhibited by incoherence in the normal state. A similar analysis confirms coherent superfluidity in the superconducting phase of Model 1 (Appendix F).

6.4 Discussion

We studied two models exhibiting superconducting transition out of an IM phase with T -linear resistivity within the framework of connected SYK “quantum dots”. By having a solvable limit exhibiting this phe- monena ubiquitous in correlated systems, we explicitly established implications of a strongly correlated incoherent normal state on superconductivity. The severe electron-electron scattering that destroys coherent quasi-particles and drives T -linear resistivity does not inhibit formation of a coherent superconducting state.

Instead, the electron-electron scattering leads to dramatic enhancement in the gap ratio, while also driving the superconducting transition first-order.

149 It is instructive to contrast the gap ratio enhancement seen in our IM-SC transition to that obtained in the standard Eliashberg theory of phonon-mediated superconductivity. Within Eliashberg theory, a relatively gentle deviation of the measured gap ratio from the universal BCS value in elemental superconductors and alloys can be accounted for [271, 272]. Such enhancement is driven by a suppression of the Tsc due to fluctuation effects ignored in the BCS mean-field theory. However, due to the large-N limit, the effects of retardation of the pairing interaction on the Fermion self-energy are suppressed, and the gap equations in (6.3) are actually exact. Thus, the enhancement of the gap ratio in our model occurs not due to the suppression of Tsc by the thermal fluctuations of the anomalous Green’s function, but rather due to the non-quasiparticle nature of the IM Green’s functions. It is worth noting that an enhancement of the gap ratio is also seen in holographic models of superconductors [273].

Interestingly, an extreme gap ratio enhancement is widely seen in various correlated-electron superconductors, such as in cuprates and iron-based superconductors [17, 44]. Our work presents the first microscopic mechanism of such an enhancement that is not driven by the suppression of Tsc by pairing fluctuations, but rather through the redistribution of spectral weight of an incoherent, non-Fermi liquid normal state.

150 Life is infinitely stranger than anything which the mind of man could invent.

Arthur Conan Doyle

7 A critical strange metal from fluctuating gauge

fields in a solvable random model

7.1 Introduction

A number of models of strange metals have been been constructed [40–42, 217, 235, 256, 257, 265, 266,

274, 275] by connecting together ‘quantum islands’, in which each island has random all-to-all interactions between the electrons i.e. each island is a 0+1 dimensional SYK model [35–37, 225]. Some of these models

[42, 217, 265, 266] exhibit ‘bad metal’ behavior above some crossover temperature, with a resistivity which increases linearly with temperature (T ), and has a magnitude (in two dimensions) which is larger than the

151 quantum unit of resistance h/e2. These models can be useful starting points for understanding a variety of experiments above moderate values of T , and they also predict [35, 217] the frequency independent density fluctuation spectrum observed in recent electron scattering experiments [229]. However, some of the most interesting and puzzling observations exhibit [11, 117, 276] linear-in-T resistivity down to vanishingly small

T , with a resistivity which is much smaller than h/e2. Kondo-like two-band SYK models have been proposed for such behavior [265, 266], in which a band of itinerant electrons acquires marginal-Fermi liquid behavior

[228] upon Kondo exchange scattering off localized electrons in SYK islands. The holographic models of strange metals have a structure very similar to these Kondo-SYK models [208, 222, 230].

A possible shortcoming of the two-band SYK-Kondo models [265, 266] is that density of itinerant carriers is ‘small’. In other words, only the itinerant electrons carry current and exhibit marginal-Fermi liquid behavior, while the localized electrons in SYK islands only act as a background ‘bath’ of incoherent electrons which dissipates current from the itinerant electrons. This behavior does not appear to be in accord with estimates of the magnitude of the linear-in-T resistivity as T → 0 [11].

In this chapter, we shall propose and solve a SYK-like model which exhibits strange metal resistivity as

T → 0, and in which the density of itinerant fermions is ‘large’. We shall examine a model of fermions coupled to an emergent, dynamic, U(1) gauge field. We shall show that a solvable SYK-like large N limit exists, in which the electrons are in N clusters with M sites per cluster (M/N is fixed as the large N limit is taken): see Fig. 7.1. The DC conductivity of our model is presented in Eq. (7.49), and the resistivity varies as

T 2x as T → 0, with the exponent x dependent only upon M/N and the particle-hole asymmetry parameter

E, as shown in Eq. (7.23) and Fig. 7.3. In the limit of small M/N, 2x ∼ 1 (see Fig. 7.3), and then we have nearly linear-in-T resistivity.

The problem of a finite density of fermions coupled to an emergent gauge field appears in many different physical contexts. The most extensively studied case is that related to compressible quantum Hall states in a half-filled Landau level [31]. These studies begin with assumption that the fermions form a Fermi surface,

152 Figure 7.1: A cartoon of our model. It consists of N clusters indexed by i, j, ..., each of which contains M sites indexed by α, β, .... Random hopping occurs between all possible pairs of intra-cluster and inter-cluster sites, but only inter-cluster hops are coupled to dynamic U(1) gauge fields Aij. The model is solved in the M,N → ∞ limit, with M/N fixed. and Landau damping from the fermions leads to an overdamped gauge propagator. The effects of the gauge coupling and the disorder are then treated perturbatively. The presence of disorder has a relatively modest effect in inducing a diffusive form for the gauge propagator. In the present chapter we shall take a random all-to-all form of the fermion propagator, and show that this allows for an exact treatment of the gauge fluctuations. The local criticality exhibited by our model is expected to eventually crossover at low enough

T to more generic finite-dimensional behavior, but there is no theory yet for such a fixed point with strong disorder and interactions.

The physical context most appropriate for our proposed connection to observations on the overdoped cuprates [11, 117] is the theory of an ‘algebraic charge liquid’ (ACL) [277] of spinless fermionic chargons coupled to an emergent gauge field. Specifically, in a SU(2) gauge theory of optimal doping quantum criticality [278–281], it has been proposed that there could be an overdoped phase with a large density of fermionic chargons coupled to a deconfined SU(2) gauge field. For simplicity, this chapter will consider the U(1) gauge field case, although the properties of the SU(2) case are expected to be very similar.

We will begin in Section 7.2 by defining the model and computing its saddle point equations in the large

153 N limit. The properties of the single fermion Green’s function as a function of frequency, temperature, and chemical potential will be described in Section 7.3. The thermodynamics will be described in Section 7.4, and we will describe a higher-dimensional generalization which allows us to compute transport properties in Section 7.5.

Appendix G.1 describes an extension of our model in which the condensation of a charge 2 Higgs field leads to a metallic phase in which the fermions carry Z2 gauge charges. The Higgs condensate quenches the gauge field fluctuations, and the transport is therefore Fermi-liquid like. The Higgs condensate also reduces the density of low-energy fermionic excitations, and so we may view this transition as a model [278–281] of optimal doping criticality from the overdoped side (no Higgs condensate) to the underdoped side (Higgs condensate present).

7.2 Model and large-N limit

We study a model of N clusters, each with M flavors of fermions, with infinite-ranged random hopping between the clusters that is coupled to fluctuating U(1) gauge fields. It is given by

∑N ∑M [ ] 1 † † H = − tαβeiAij f f + (MN)1/2µδαβf f , (MN)1/2 ij iα jβ ij iα iα ij=1 αβ=1

≪ αβ βα ≫ ≪ | αβ|2 ≫ 2 − tij tji = tij = t ,Aji = Aij. (7.1)

→ ∞ O αβ where N,M and M/N is an (1) quantity. The tij are complex gaussian random variables and

≪ .. ≫ denotes disorder-averaging; all disorder averages other than the ones explicitly shown above are zero. The clusters are indexed by i, and the sites (flavors) within a cluster, are indexed by α. A cartoon of our model is shown in Fig. 7.1.

As in the analysis of the SYK models [37, 41], we average over realizations of disorder. Doing so formally

154 requires introducing replicas; however we assume, like in the SYK models, that there is no replica-symmetry breaking, restricting to replica-diagonal configurations and suppressing the then trivial sum over replicas. We introduce bilocal (in time) fields G and Σ, obtaining the Euclidean action

∫ ∑N ∑M † 0 S = dτ fiα(τ)(∂τ + iAi (τ) + µ)fiα(τ) i=1 α=1 ∫ ∑N M ′ 2 ′ i(Aij (τ)−Aij (τ )) ′ ′ + t dτdτ e Gj(τ − τ )Gi(τ − τ) N ≤ ij=1,i j [ ] ∫ ∑N ∑M 1 † − M dτdτ ′ Σ (τ − τ ′) G (τ ′ − τ) − f (τ ′)f (τ) . (7.2) i i M iα iα i=1 α=1

∫ The partition function is given by Z = DfDf †DADGDΣ e−S, and τ denotes Euclidean time. Unbounded integrals denote integration over the full range of the pertinent variable. Integrating out the Lagrange multi- pliers Σi followed by the Gi restores the pure disorder-averaged action. In the M → ∞ limit, the integrals over the Σi enforce the definitions of Gi on each cluster i. The disorder averaged action is gauge-invariant under the transformations

→ − → iθi(τ) 0 → 0 − Aij(τ) Aij(τ) + θi(τ) θj(τ), fiα(τ) fiα(τ)e ,Ai (τ) Ai (τ) ∂τ θi(τ), (7.3)

′ ′ i(θ (τ)−θ (τ ′)) ′ ′ i(θ (τ)−θ (τ ′)) with Gi(τ − τ ) → Gi(τ − τ )e i i and Σi(τ − τ ) → Σi(τ − τ )e i i . The propagators

0 0 of the scalar potentials Ai will be screened due to the finite density of fermions [27]; fluctuations of the Ai will be hence unable to inflict any singular self energy on the fermions at low energies, and we will thus

0 simply ignore the Ai .

Examining the disorder-averaged action, after integrating out the fermions, does not immediately suggest a large-N saddle-point for the Gi, but a simple large N limit does turn out to exist. The reason is that there are enough (M) sites per cluster to self-average the cluster Green’s function Gi, so that the solution will have Gi that don’t depend on i, even though there are N clusters. This can be seen easily when the coupling

155 to the gauge fields is turned off. Then we know the standard result for the fully-averaged Green’s function

Gavg of the full large-MN random matrix exactly, but can also express it as

∑N ∑M ∑ 1 † 1 G (τ − τ ′) = ⟨f (τ)f (τ ′)⟩ = G (τ − τ ′). (7.4) avg MN iα iα N i i=1 α=1 i

Then, the second term of (7.2) may be written as

∫ t2 ∑N M dτdτ ′G (τ − τ ′) G (τ ′ − τ). (7.5) 2 avg i i=1

Since there are now appropriate prefactors of M everywhere in all terms in S after integrating out the fermions, we can take functional derivatives with respect to Gi and Σi (remembering that Gavg contains

′ 2 ′ Gi) and write down the saddle-point Σi(τ − τ ) = t Gavg(τ − τ ) and Gi(iωn) = 1/(iωn + µ − Σi(iωn)), which are independent of i, indicating that the cluster-averaged (over M sites) Green’s function is the same as the fully averaged (over MN sites and clusters) Green’s function at large-M,N. Another way to see this qualitatively is that the distribution for G’s averaged over M sites is the convolution of M distributions for the single-site G’s. For Gaussians, this would imply that its variance is 1/M th of that of the single-site distribution, which, although much larger than the variance of the fully averaged G (which is 1/(MN)th of that of the single-site distribution), should still be small as M → ∞.

Turning the gauge fields back on, we expand out the exponentials to quadratic order (assuming that monopoles are irrelevant and there is no confinement transition, so the compactness of the gauge fields

156 isn’t important; we will discuss this further at the end of Sec. 7.4) and obtain,

∫ ∑N ∑M † 0 S = dτ fiα(τ)(∂τ + iAi (τ) + µ)fiα(τ) i=1 α=1 [ ] ∫ ∑N 2 M ′ − ′ − 1 2 − 1 2 ′ ′ + t dτdτ 1 + i(Aij(τ) Aij(τ )) Aij(τ) Aij(τ ) + Aij(τ)Aij(τ ) N ≤ 2 2 ij=1,i j [ ] ∫ ∑N ∑M 1 † × G (τ − τ ′)G (τ ′ − τ) − M dτdτ ′ Σ (τ − τ ′) G (τ ′ − τ) − f (τ ′)f (τ) . (7.6) j i i i M iα iα i=1 α=1

This expanded-out action is also gauge-invariant under the previously mentioned transformation, up to quadratic order in the gauge fields and their shifts. The terms linear in Aij in the second line of the above

2 vanish, and the Aij terms can be reorganized,

∫ ∑N ∑M † 0 S = dτ fiα(τ)(∂τ + iAi (τ) + µ)fiα(τ) i=1 α=1 T ∑ ∑N + A (iΩ ) [Π (iΩ ) − Π (iΩ = 0)] A (−iΩ ) 2 ij m ij m ij m ij m Ωm ij=1,i≤j ∫ ∑N 2 M ′ ′ ′ + t dτdτ Gj(τ − τ )Gi(τ − τ) N ≤ ij=1,i j [ ] ∫ ∑N ∑M 1 † − M dτdτ ′ Σ (τ − τ ′) G (τ ′ − τ) − f (τ ′)f (τ) , (7.7) i i M iα iα i=1 α=1 with ∫ M Π (iΩ ) = 2t2 dτeiΩmτ G (τ)G (−τ). (7.8) ij m N i j

We proceed to integrate out the fermions and the gauge fields. Normally, integrating out the gauge fields requires gauge-fixing in order to avoid overcounting redundant configurations. However, in the large-N

2 limit here, we have O(N ) gauge variables Aij, but only O(N) constraining variables θi. The space of

2 gauge field configurations is then ∼ RN , whereas the space occupied by configurations redundant to a

2 N single configuration, generated by shifting the O(N ) Aij’s by N θi’s is ∼ R . Therefore the space of

157 2 2 unique gauge configurations is ∼ RN /RN , which at leading order in large-N is approximately RN . Thus, we can just naively integrate out the Aij in the large-N limit, and the corrections from gauge-fixing will not affect the free energy and the saddle-point values of G and Σ at leading order in the large-N limit. After integrating out, we obtain

∑ ∑N T ∑ ∑N TS = −MT ln [iω + µ − Σ (iω )] + ln [Π (iΩ ) − Π (iΩ = 0)] n i n 2 ij m ij m ωn i=1 Ωm=0̸ ij=1,i

∑ − 2 2 G(iωn + iΩm) G(iωn) Σ(iωn) = t G(iωn) + t T , Π(iΩm) − Π(iΩm = 0) Ωm=0̸ ∑ 2 M 1 Π(iΩm) = 2t T G(iωn)G(iωn + iΩm),G(iωn) = . (7.10) N iωn + µ − Σ(iωn) ωn

These equations can also be derived diagrammatically starting from (7.1) in the large-M,N limit, and expanding the exponential to quadratic order after disorder-averaging (Fig. 7.2).

Note that the zero Matsubara frequency component of Aij does not contribute to the action (7.7) or (7.9) even at T ≠ 0. The gauge field contribution to the fermion self energy Σ(iωn) in (7.10) thus doesn’t involve the zero Matsubara frequency component of the gauge field propagator. This is because, as far as the fermions are concerned, the zero Matsubara frequency components are just static phase shifts of the

αβ tij , and have already been accounted for while disorder averaging. This absence of the zero frequency components has consequences for the thermodynamic properties of the saddle-point solution, and certain modifications have to be made to ensure that the saddle-point is thermodynamically stable (see Sec. 7.4).

158 Figure 7.2: Diagrammatic representation of the fermion (Σ) and regularized gauge field (Π˜ = Π(iΩm)−Π(iΩm = 0)) self-energies for the Dyson equation (7.10). The black lines are fermion propagators, the red lines are gauge field αβ propagators, and the dashed blue lines are contractions of the gaussian random variables tij coming from the disorder average. These are the only diagrams that contribute in in the large-M,N limit.

However, these modifications do not affect the saddle-point solution to be detailed in the next section above some energy scale which can be made arbitrarily small.

If we consider fluctuations (δGi(iωn), δΣi(iωn)) about the saddle-point action that do not amount to

(1) (2) simply changing a gauge, the kernel of their action at quadratic order is given by Kˆij = Kˆ δij + Kˆ , where Kˆ (1,2) are matrices in (δG, δΣ) and frequency space. Here Kˆ (1) is of order M, coming from the fermion determinant and ΣG terms of (7.9), and Kˆ (2), which comes from the other two terms is of order 1.

Then, diagonalizing Kˆ in i, j and (δG, δΣ) space produces O(N) fluctuation eigenmodes with eigenvalues that are O(M). Integrating over these N modes yields a sub-leading O(N) contribution to the free energy, and each of these modes also has an O(M) stiffness that suppresses its fluctuations. Hence, the saddle-point described by (7.10) is well-defined.

159 7.3 Single-particle properties

7.3.1 Zero temperature

We solve for the fermion and gauge field propagators at T = 0. We set µ = 0 (corresponding to half-filling, see Sec. 7.3.2 for µ ≠ 0), and start with an ansatz for G in the IR at T = 0,

( ) sgn(τ) πx 1 G(τ) = −C ,G(iω ) = −2iCtx−1 sin Γ(x)sgn(ω )|ω |−x, 0 < x < , C > 0. t1−x|τ|1−x n 2 n n 2 (7.11)

We then obtain

2 2x 1−2x Π(iΩm) − Π(iΩm = 0) = −4(M/N)C t sin(πx)Γ(2x − 1) |Ωm| . (7.12)

Thus the fermion self-energy

√ ( ) 2x−1 πx 2 iN π2 sin 2 csc (πx) 1−x x Σ(iωn) = ( ) sgn(ωn)t |ωn| 2MCxΓ(2x − 1)Γ 1 − x [ ∫ 2 ] 2 dΩm 1 + t G(iωn) 1 − . (7.13) 2π Π(iΩm) − Π(iΩm = 0)

The integral over Ωm contains contributions from frequencies outside the regime of validity of the IR solution, and hence requires a UV completion in order to be evaluated. We assume that the UV completion is such that the term in square brackets evaluates to zero, which we will justify below; the vanishing of the square bracketed term is also confirmed by our numerical analysis of the UV complete theory below. Then, using

G(iωn) = −1/Σ(iωn), we find that we cannot determine C (it cancels between the LHS and RHS of the

160 0.5

0.4

0.3

0.2

0.1

0.0 10-4 0.01 1 100

Figure 7.3: Plot of the exponent x giving the frequency scaling of the IR fermion self-energy, vs 2M/N, at half-filling. equation), but we can determine the universal exponent x by solving

1/x − 2 2M = , (7.14) 1 + sec(πx) N with x vs 2M/N plotted in Fig. 7.3. The fact that we can’t determine C purely from the IR properties indicates that it is non-universal.

We now justify the vanishing of the term in square brackets in (7.13): Suppose it didn’t exactly vanish, ∫ 2 and dΩm/(Π(iΩm)−Π(iΩm = 0)) = 1−ν, where ν ≪ 1. Then, this leaves behind a term νt G(iωn) in

−x the expression for Σ(iωn), which, scaling as sgn(ωn)|ωn| , is more relevant at low energies than the other term in Σ(iωn). We can then try to ignore the other term in the IR. The Dyson equation becomes

2 1 Σ(iωn) = νt G(iωn),G(iωn) = , (7.15) iωn − Σ(iωn)

1/2 This equation is solved in the IR by the random-matrix solution G(iωn) = −isgn(ωn)/(ν t).

This solution then modifies the gauge field propagator in the IR, with Πnew(iΩm) − Πnew(iΩm = 0) =

161 (2/π)(M/N)(|Ωm|/ν). We can then write using (7.10)

∫ Λ − 1/2 1/2 dΩm sgn(ωn) sgn(ωn + Ωm) Σ(iωn) = −i(t/ν )sgn(ωn) + iν t , (7.16) −Λ 2π (2/π)(M/N)|Ωm| where Λ is a “critical window” over which the IR solution is valid. This then gives a singular self-energy

( ( )) ( ) Nν Nν |ω | |ω | 2M Σ(iω ) = −i(t/ν1/2)sgn(ω ) 1 + ln n → −i(t/ν1/2)sgn(ω ) n , (7.17) n n 2M Λ n Λ

We have thus recovered a power-law self-energy without explicitly assuming ν = 0 to begin with. Repeated iterations of (7.10) then converge the exponent of the power-law to the value defined by (7.14).

The Dyson equations (7.10) are not fully UV-complete, and do not contain enough information to determine the gauge field propagator at high frequencies. In order to solve them numerically, we need a

UV-complete set of equations. We do this by adding a “Maxwell” term to the gauge field action

∫ 1 ∑N S → S + dτ (∂ A (τ) + A0(τ) − A0(τ))2, (7.18) 2g2 τ ij i j ij=1,i≤j

0 with a gauge coupling g, and the Ai ’s may be ignored due to the aforementioned screening. This then adds a

2 2 − term Ωm/g to Π(iΩm) Π(iΩm = 0) in (7.10). Note that (7.18) contains only “electric” kinetic terms for the gauge fields and no “magnetic” terms that are functions of the sums of gauge link variables Aij around closed loops. We will discuss the effects of adding magnetic terms in Sec. 7.4.

The numerical solution was then performed by starting with free fermion and gauge field Green’s functions

1 g2 G0(iωn) = ,D0(iΩm) = 2 , (7.19) iωn + µ Ωm

2 and then iterating the Dyson equations (7.10) in the MATLAB code gd.m [282]. We found that the t G(iωn) term in Σ(iωn) indeed cancels out as T → 0, and a power-law scaling of G(iωn) is obtained in the IR, with

162 2 the exponent given by (7.14). This cancellation of the t G(iωn) term holds even for µ ≠ 0, leading to the results in Sec. 7.3.2.A T = 0 numerical solution of the real time version of the Dyson equations (performed in gdrealtime0.m [283]) also yields the appropriate analytically continued version of (7.11) for the retarded

Green’s function in the IR,

( ) πx GR(ω) = −2iCtx−1 sin Γ(x)(−iω)−x. (7.20) 2

At the saddle point, we have the effective action for the fluctuations of the Aij fields

A SSP ∫ [ ∫ ] M ∑N = t2 dτdτ ′ A (τ) G(τ − τ ′)G(τ ′ − τ) − δ(τ − τ ′) dτ ′′G(τ − τ ′′)G(τ ′′ − τ) A (τ ′) N ij ij ij=1,i≤j

(7.21)

x−1 −x Under the scaling τ → bτ, we have G(τ) → b G(τ) from (7.11), which then implies Aij(τ) → b Aij(τ)

− ′ from (7.21). Corrections to (7.21) coming from the expansion of ei(Aij (τ) Aij (τ )) beyond quadratic order ∫ ′ ′ n>2 ′ ′ in (7.2) are of the form dτdτ cn>2(Aij(τ) − Aij(τ )) G(τ − τ )G(τ − τ). The above scaling then

(n−2)x implies that c(n>2) → b c(n>2), so these terms are irrelevant, and their coefficients become small in the IR as b → 0, allowing us to ignore them.

7.3.2 Deviations from half-filling

For µ ≠ 0, the IR Green’s function develops a spectral asymmetry, with G(−τ < 0) = −e−2πE G(τ > 0) at T = 0; C(E) C(E)e−2πE G(τ > 0) = − ,G(τ < 0) = . (7.22) t1−xτ 1−x t1−x|τ|1−x

163 The polarization Πij(τ) and the gauge field propagator however remain symmetric about τ = 0. The real part of the self energy satisfies Re[Σ(iωn → 0)] = µ, cancelling the chemical potential in the Green’s

2 function. The t G(iωn) term in the self-energy Σ(iωn) still cancels out as before. However, interestingly, the exponent x of the power-law scaling depends on the asymmetry parameter E and is given by the solution to (1/x − 2)(cosh(2πE) − cos(πx)) 2M = . (7.23) tan(πx) sin(πx) N

This relation can be determined from the Dyson equation (7.10) following Ref. [35], and gives x → 1/2 as

E → ∞ regardless of M/N.

As in the SYK models [37, 41], the relationship between E and µ is nonuniversal, and depends on the values of UV details. However, following Ref. [244], a universal relationship between the asymmetry parameter and the filling can be determined: The filling q0 can be written as

∫ ∞ dE F iE0+ q0 = i G (E)e , (7.24) −∞ 2π where ∫ ∞ ρ (E ) F ≡ dE1 f 1 G (E) + (7.25) −∞ 2π E1 − E − i0 sgn(E1)

R is the Feynman Green’s function, with ρf (E1) ≡ −2Im[G (E1)] the fermion spectral function. As in

Ref. [244], we have

∫ ∫ ∞ R ∞ 1 R − − R −∞ P dE ∂EG (E) iE0+ − P dE F F iE0+ q0 = (arg[G (0 )] arg[G ( )])+i R e i G (E)∂EΣ (E)e , π −∞ 2π G (E) −∞ 2π (7.26)

R where P denotes the Cauchy principal value. Obtaining the low-energy forms of G (E) and hence ρf (E)

164 from (7.22), and using GR(E → ∞) = 1/(E + i0+), this can then be written as

∫ ∞ x −iπx/2 dE F F iE0+ q0 = 1 + + arg[−i sin(π(x/2 − iE))e ] − iP G (E)∂EΣ (E)e . (7.27) 2 −∞ 2π

The remaining integral needs to be computed carefully using the Dyson equation (7.10) and the methods described in Appendix A of Ref. [244]. We obtain

x 2Nκ(x)Γ2(x) sin(2πx) sinh(2πE) q = 1 + + arg[−i sin(π(x/2 − iE))e−iπx/2] − , (7.28) 0 2 MΓ(2x − 1)

∑ 6 I where κ(x) = i=1 i(x), with

cot(πx)Γ(1 − x)Γ(2x) csc(πx)Γ(2x)(γ + ψ(1 + x)) I = − , I = − E , 1 16π2Γ(1 + x) 2 16π2Γ(x)Γ(1 + x)

∫ ∫ iπx −1 0 −x − 2x−1 I − e Y1 ( Y2) 3 = 3 dY1 dY2 , 16π −∞ −∞ (Y1 + Y2)(1 + Y1 + Y2) ( ( ) ∫ Y1 0 2F1 1, 1 − 2x; 2 − 2x; I 1 − −x 2x−1 − Y1+1 4 = 3 dY1( Y1) (Y1 + 1) 16π − 2x − 1 1 )

− ln(Y1 + 1) + γE + ψ(1 − 2x) , ∫ ∫ iπx − 0 0 − − − I −e xΓ(1 x) θ( Y1 Y2 1) −x − 2x−1 5 = 3 dY1 dY2 2 Y1 ( Y2) 16π Γ(2 − x) −∞ −∞ (Y1 + Y2) ( ) 1 × 2F1 1, 1 − x; 2 − x; − , Y1 + Y2 ∫ ∫ iπx 0 0 I − e x −x − 2x−1 − 6 = 3 dY1 dY2 θ(Y1 + Y2 + 1)Y1 ( Y2) 2F1 (1, 1 + x; 2 + x; (Y1 + Y2)) , 16π (1 + x) −∞ −∞

(7.29)

where ψ is the digamma function, θ is the Heaviside step function, 2F1 is a hypergeometric function [284],

2 and γE is the Euler-Mascheroni constant. κ(x → 1/2) = −1/(16π), and κ(x → 0) ∝ −1/x (see

165 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 -2 -1 0 1 2 0.1 0.2 0.3 0.4 0.5

Figure 7.4: (a) Plot of the filling q0 vs the asymmetry parameter E for 2M = N. (b) Plot of the function −κ(x) defined in (7.29) vs the self-energy exponent x.

Fig. 7.4b). Putting together (7.23), (7.28) and (7.29), we see that q0 is a smooth function of E that decreases monotonically from 1 to 0 as E is swept from −∞ to ∞ (see Fig. 7.4a). This dependence of q0 on E also agrees quantitatively with that obtained from the numerical solutions of (7.10), in which q0 is given by

− −2πE R − R + q0 = G(τ = 0 ) and e = Im[G (ω = 0 )]/Im[G (ω = 0 )].

7.3.3 Nonzero temperature

The regularized IR Dyson equations (7.10) can be written in the time domain using two-time notation as (see

Ref. [37])

∫ 2 2 − dτ3G(τ1, τ3)Σ(˜ τ3, τ2) = δ(τ1, τ2), Σ(˜ τ1, τ2) = t νG(τ1, τ2) + t G(τ1, τ2)D(τ2, τ1) ∫

dτ3D(τ1, τ3)Π(˜ τ3, τ2) = δ˜(τ1, τ2), ( ∫ ) M Π(˜ τ , τ ) = 2t2 G(τ , τ )G(τ , τ ) − δ(τ , τ ) dτ G(τ , τ )G(τ , τ ) , (7.30) 1 2 N 1 2 2 1 1 2 3 1 3 3 2 where

ν = 1 − lim[D(τ1, τ2) + DUV(τ1, τ2)], (7.31) 2→1

166 ˜ − −1 −1 and δ(τ1, τ2) = δ(τ1, τ2) Lτ , where Lτ is the length of the time domain (Lτ = T at a finite temperature

T ). The chemical potential µ has been absorbed into Σ to regularize it to Σ˜. We split the gauge field propagator into an IR piece D and a UV piece DUV. The UV piece is not determined by (7.30), and is not sensitive to rescalings of τ. The reason that the δ˜ appears instead of just a δ is because the action (7.7) doesn’t contain zero frequency modes of Aij. As a result, D here doesn’t contain a zero frequency mode either, and consequently the pertinent delta function should be modified to remove its zero frequency mode.

On a time domain of infinite size (such as at zero temperature), the zero frequency mode occupies a measure zero subspace, and then there is no difference between δ˜ and δ.

The equations (7.30) are not invariant under a general set of reparametrizations with τ = f(σ) and an arbitrary function h,[37]

→ h(σ1)/h(σ2) ˜ → h(σ1)/h(σ2) ˜ G(τ1, τ2) ′ ′ a G(σ1, σ2), Σ(τ1, τ2) ′ ′ 1−a Σ(σ1, σ2), (f (σ1)f (σ2)) (f (σ1)f (σ2))

′ ′ 2a−1 ′ ′ −2a D(τ1, τ2) → (f (σ1)f (σ2)) D(σ1, σ2), Π(˜ τ1, τ2) → (f (σ1)f (σ2)) Π(˜ σ1, σ2), (7.32)

˜ −1 because of the second term in the expression for Π, and additionally because ν and Lτ can be nonzero.

However, they can still be scale invariant under τ → bτ iff

G → b−2aG, Σ˜ → b2a−2Σ˜,D → b4a−2D, Π˜ → b−4aΠ˜, ν → b4a−2ν. (7.33)

Note that a is not determined by these equations, but we choose 2a = 1 − x due to the particular power-law scaling of Sec. 7.3.1 that is selected when the UV-complete equations are solved.

Now consider applying the scale transformation at a finite temperature. Since τ ∈ [0, 1/T ), this also scales T → T/b, leaving T τ invariant. (7.30) is then compatible with a scaling solution (reverting back to

1−x 2x one-time notation) G(τ) ∝ T FG(τT ) (and corresponding expressions for D, Σ˜ and Π˜) iff ν ∝ T . To check that we indeed get this behavior of ν, we use the definition (7.31) of ν, the fact that DUV is not affected

167 6.5 1.5 6

5.5 1.4

5 1.3 4.5 ● 4 ● ● ● ● 1.2 3.5

3 1.1

2.5 1.0 2 10-6 10-5 10-4 0.001 0.010 0 10 20 30 40 50 60 70 80 90 100

x Figure 7.5: (a) Plot of the scaling form of the fermion self-energy −Im[Σ(iωn)]/T vs ωn/(πT ) obtained by numerical solution of the imaginary-time Dyson equations for different values of T . For all curves, t = g2 = 1, 2M = N and µ = 0 (corresponding to x = 0.230651 from (7.14)). The curves collapse onto one another at low frequencies, confirming a universal low-energy scaling form. The deviations from universality at higher frequencies and temperatures occur because of the finite size of the critical window Λ over which the low-energy solution is valid. (b) Plot of the quantity ν(T )/T 2x vs T for the same values of parameters. This clearly shows ν(T ) ∝ T 2x over a large range of temperatures.

2x by rescalings of T at low T ≪ Λ, and the scaling form for D(τ) ∝ T FD(τT ), to obtain

2x 2x ν(T ) − ν(0) = lim lim[T FD(τT )] − lim T FD(τT ), (7.34) T →0 τ→0 τ→0 which gives ν(T ) ∝ T 2x when ν(0) = 0, which we already established in Sec. 7.3.1. Thus, the low- energy Dyson equations in the gauge field problem are fully consistent with a scaling solution at small finite temperatures. Our numerical solution confirms this (Fig. 7.5a), and we also find ν(T ) ∼ T 2x numerically at small T (Fig. 7.5b).

The IR fermion Green’s function in the gauge field case does not have a conformally remapped form at

T ≠ 0, as the equations (7.30) are not invariant under (7.32) with τ = tan(πT σ)/T , but instead obeys

( ) C(E) ω 1 G(iω ,T ) = F n , E ,F (y → 0, E) ∝ y0,F (y → ∞, E) ∝ . (7.35) n t1−xT x G T G G yx

We can only compute the scaling function FG numerically. The self-energy also satisfies a low-energy scal-

−1 1−x x ing form Im[Σ(iωn)] = −C (E)t T FΣ(ωn/T, E). However, the scaling function FΣ again differs

168 Ratio T = 10−4 T = 10−5 T = 10−6 Conformal n31 1.3680 1.3703 1.3709 1.2607 n53 1.1363 1.1382 1.1387 1.1224 n75 1.0845 1.0862 1.0865 1.0800 n97 1.0611 1.0626 1.0629 1.0594

Table 7.1: Comparision of numerical values of ratios nab ≡ Im[Σ(iaπT )]/Im[Σ(ibπT )] in the gauge field problem at different temperatures with those derived from the conformal Green’s function Gc(τ) = −(Cπ1−x/t1−x)(T/ sin(πT |τ|))1−x). As T is reduced, these ratios converge to universal values that differ significantly from the conformal ones at low energies, which implies that the scaling function FΣ (or FG) is not the conformal one, and that the local criticality in the gauge field model is different from the SYK universality class. The values of other parameters used are the same as those used in Fig. 7.5, but these universal low-energy ratios are insensitive to the values of t and g2 as T → 0 within numerical tolerances.

from the conformal scaling function for the same exponent x (corresponding to the self-energy Σc(iωn) =

1−x 1−x 1−x −1/Gc(iωn) derived from the conformal Green’s function Gc(τ) = −(Cπ /t )(T/ sin(πT |τ|)) ), as can be seen by comparing universal ratios such as n31 ≡ Im[Σ(3iπT )]/Im[Σ(iπT )] with their corresponding conformal values (see Table 7.1).

7.4 Thermodynamics

In this section we describe the thermodynamic properties of the saddle-point solution described in the previous two sections at low temperatures. We specialize to the case of half-filling, with µ = 0. This allows for temperature derivatives of the free energy at a constant fermion density of half-filling to be the same as its temperature derivatives at constant zero chemical potential, which are easier to evaluate. We do not expect any of the qualitative features discussed here to be modified away from half-filling.

The free energy can be written down from (7.9) evaluated at the saddle-point. It is

∑ [ ] ∑ iωn 2 MN 2 F = −T ln Z = MNT ln − MNT ln 2 + t T G (iωn) iωn − Σi(iωn) 2 ωn ωn ∑ N 2T ∑ − MNT Σ(iω )G(iω ) + ln [Π(iΩ ) − Π(iΩ = 0)] , (7.36) n n 4 m m ωn Ωm=0̸

169 where we added and subtracted the free fermion contribution, so that the frequency sum involving the logarithm converges and we may evaluate it numerically. The term on the second lime represents the gauge field contribution. Setting this aside for the moment, and numerically evaluating the fermion contribution using the saddle point of the UV complete action with the electric “Maxwell” term (7.18), we obtain

Ff /(MN) ≈ −c0(M/N) + T c1(M/N) as T → 0. This implies that the fermion contribution to the spe-

2 2 cific heat Cf = −T ∂ Ff /∂T vanishes at low temperatures and the fermions make a constant negative contribution Sf = −∂Ff /∂T to the total entropy. Since the fermions and gauge fields are highly entangled, we of course need to add the gauge field contribution to obtain the full physical free energy and associated thermodynamic quantities. We can write

[ ] 2 ∑ N T Π(iΩm) − Π(iΩm = 0) FA = ln 4 Π(iΩm,T = 0) − Π(iΩm = 0,T = 0) Ωm=0̸ N 2T ∑ + ln [Π(iΩ ,T = 0) − Π(iΩ = 0,T = 0)] , (7.37) 4 m m Ωm=0̸ where we have added and subtracted a term that is evaluated using the zero temperature functional form of

Π, but evaluated at the Matsubara frequencies corresponding to a particular temperature. Since Π(iΩm) −

Π(iΩm = 0) obeys a quantum-critical scaling form, the first term becomes ∑ F (1) − NT 1−2x A /(MN) = 4M m̸=0 ln[(2πm) FD(2πm)].

1−2x 2 We find numerically that FD(2πm) = 1/(2πm) + O(1/m ) for m ≫ 1 in the scaling limit, so this

F (1) ∼ − sum converges at large m, and leads to A /(MN) T , which doesn’t contribute to the specific heat and provides a constant contribution to the entropy at low temperatures.

170 The second term of (7.37) can be computed by zeta-function regularization using the result (7.12):

[ ] (2) ∑ F NT M − A = ln −4 C2t2x sin(πx)Γ(2x − 1) |2πm|1 2x T 1−2x MN 4M N m=0̸ [ ] NT M = − ln −4 C2t2x sin(πx)Γ(2x − 1)T 1−2x . (7.38) 4M N

This produces the dominant contribution to the low-temperature specific heat,

C C(2) ∂2F (2) (1 − 2x)N ≈ A = −T A = , (7.39) MN MN ∂T 2 4M which is positive and extensive. In the limit of M/N → ∞, where x → 0 and the non-Fermi liquid solution turns into a noninteracting random matrix solution, this large contribution to the specific heat vanishes as it √ should, and in the opposite limit of M/N → 0, where x → 1/2, it blows up as 1/ M/N, as can be seen by applying (7.14).

F (2) The free energy contribution A also leads to the dominant contribution to the low-temperature entropy

S(2) − F (2) ∝ − A /(MN) = ∂ A /∂T (1 2x)(N/(4M)) ln T . This is negative at low T , which indicates that our theory is incomplete: extra degrees of freedom must be present in a physical theory in order to offset this entropy. The reason this happens is that our theory is missing all information about the zero-frequency modes of the Aij. In any sensible electromagnetic lattice gauge theory, these modes will contribute to physical static magnetic field configurations that cost energy: Exciting a single link Aij will lead to nonzero magnetic fluxes through all plaquettes containing that link, and a magnetic “Maxwell” term acting on these fluxes will contribute to the action, even if they are static. However such terms are not generated in our theory by integrating out the fermions in the large-N,M limits. In order to generate these terms we need to appeal to some heavy degrees of freedom that couple to the gauge fields in such a way that integrating out these degrees of freedom will produce magnetic “Maxwell” terms.

171 Assuming this is the case, we write down the simplest possible gauge and time-reversal invariant magnetic

“Maxwell” action that is appropriate for an all-to-all interacting theory without any spatial structure. It is

∫ m2 ∑ S = B dτ (A (τ) + A (τ) + A (τ))2, (7.40) B 2(N − 2) ij jk ki △ijk where the sum runs over all possible unique triangles. The kernel of this quadratic action has (N − 1)(N −

2 − − 2)/2 degenerate eigenvectors with eigenvalue mB(1 + 2/(N 2)) and N 1 degenerate eigenvectors with eigenvalue 0. The zero-eigenvalued eigenvectors are all pure gauge and can each be gauge-transformed to the configuration Aij = 0; they correspond to the state in which the flux through all triangles is zero, and thus do not contribute anything to the free energy. In the large-N limit, the thermodynamic fraction of modes residing on a single links Aij have negligible overlap with the zero-eigenvalued eigenvectors. This permits the approximation, exact in the infinite-N limit

∫ m2 ∑N S ≈ B dτ A2 (τ). (7.41) B 2 ij ij=1,i≤j

2 − 2 − 2x 1−2x We assume that mB is much smaller than 4(M/N)C sin(πx)Γ(2x 1)t T . Then including this term just adds N 2 F (3) = T ln m2 (7.42) A 4 B to (7.36). This term doesn’t contribute to the specific heat, but offsets the leading contribution to the entropy

S(2)+(3) − 2 − 2x 1−2x 2 to a large positive value A /(MN) = (N/(4M)) ln( 4(M/N)C sin(πx)Γ(2x 1)t T /mB).

− 2 − 2x 1−2x ≪ 2 For 4(M/N)C sin(πx)Γ(2x 1)t T mB, the fermions effectively end up coupling to gapped

172 bosonic modes. The low-energy Dyson equation then reads

  ∑ 2  − 1  Σ(iωn) = t G(iωn) 1 T 2 m + Π(iΩm) − Π(iΩm = 0) Ωm=0̸ B ∑ 2 G(iωn + iΩm) + t T 2 , m + Π(iΩm) − Π(iΩm = 0) Ωm=0̸ B ∑ 2 M 1 Π(iΩm) = 2t T G(iωn)G(iωn + iΩm),G(iωn) = . (7.43) N iωn + µ − Σ(iωn) ωn

The term in square brackets no longer cancels at T = 0, as increasing the denominator of the boson propaga-

2 tor by adding a mass makes its value smaller than the zero-mass case. This leaves behind a νt G(iωn) term in Σ(iωn), leading to a renormalized random-matrix solution at the lowest energies. The second term in the first line of (7.43) vanishes at small external frequencies ωn, as G(iΩm) is odd in Ωm and the denominator is a constant at low frequencies (for nonzero chemical potential, this sum just produces a constant that is absorbed by µ). These points can be easily verified by numerically solving the UV-completed version of

(7.43) using the MATLAB code gd.m [282]. The lowest energy state then has a vanishing entropy and specific

2 heat. Henceforth, we shall assume that we are only interested in energy scales larger than the small mB, treating it as an IR regulator much smaller than T , and focus on the non-Fermi liquid.

− We also checked numerically that the compressibility MN∂q0/∂µ|T , where q0 = G(τ = 0 ) asymptotes

→ 0 to a nonzero constant as T 0. This justifies our rationale of ignoring the time components Ai of the gauge fields in the IR, as their propagators are screened by this compressibility.

Finally, from the point of view of the magnetic “Maxwell” terms, the model behaves like a U(1) gauge theory in a large (O(N)) number of dimensions. Possible magnetic monopoles arising due to the compactness of the U(1) gauge group then source nonzero fluxes through a large number of plaquettes, leading to

O(N) increases in the free energy through the magnetic Maxwell terms, while not coupling to the fermions by virtue of being a static background. Thus, the configuration in which no monopoles exist should be a

173 stable saddle-point, and monopole operators are irrelevant.

7.5 Transport

In order to consider transport properties of this model, we need to make appropriate modifications. First, we need some spatial structure. This can be achieved by defining the clusters indexed by i, j to lie on the sites of an N-dimensional hypercubic lattice, with each cluster then having 2N neighbors. The fermions hop between nearest neighbor clusters, coupling to gauge fields Aij that live on the bonds of the lattice.

Second, for an external probe gauge field to drive a current, it must couple to a different charge from the one that the internal gauge fields Aij couple to: If they coupled to the same charge, then turning on the probe field only amounts to shifting the values of Aij, and the path integral over Aij trivially absorbs these shifts, rendering the partition function immune to the probe field. If we view the fermions as chargons arising from fractionalization in an ACL, we can divide the flavors indexed by α, β into equal fractions of two species that couple to the internal gauge field with opposite charges, but which couple to the external probe gauge field with equal charges, which is a single-axis version of the SU(2) case discussed in Ref. [29]. Then, our modified version of (7.1) reads

∑ ∑M ∑ [ ] ′ 1 αβ † iA σz 1/2 αβ † H = − t f e ij ss′ f ′ + (2MN) µδ δ ′ f f , (2MN)1/2 ij iαs jβs ij ss iαs iαs ⟨ij⟩ αβ=1 ss′=

≪ αβ βα ≫ ≪ | αβ|2 ≫ 2 tij tji = tij = t . (7.44)

∑ iθ (τ)σz → i ss′ ′ → − This has a U(1) gauge invariance under fiαs(τ) s′= e fiαs (τ) and Aij(τ) Aij(τ)+θi(τ)

θj(τ).

174 Performing the same manipulations as before, we obtain

∫ ∑ ∑M ∑ ′ † 0 S = dτ fiαs(τ)(∂τ + isAi (τ) + µ)fiαs(τ) i α=1 s= ∫ ∑ ∑ [( ) ] 2 M ′ 1 2 1 2 ′ ′ ′ z + t dτdτ 1 − A (τ) − A (τ ) + A (τ)A (τ ) δ ′ + i(A (τ) − A (τ ))σ ′ 2N 2 ij 2 ij ij ij ss ij ij ss ⟨ij⟩ ss′= [ ] ∫ ∑ ∑ ∑M ′ ′ ′ ′ ′ 1 ′ † × G ′ (τ − τ )G (τ − τ) − M dτdτ Σ (τ − τ ) G (τ − τ) − f (τ )f (τ) , js is is is M iαs iαs i s= α=1

(7.45) as before, the time integrations kill the term proportional to σz in the second line of the above. This action then leads to a saddle-point symmetric in s described by (7.10), with the IR solution (7.11). Similar arguments for invariance under gauge-fixing at large-N and stability of the saddle-point as before apply.

z → z ′ We now perturb the action (7.45) with a diagonal probe field, so that Aij(τ)σss′ Aij(τ)σss′ +Ξij(τ)δss where Ξij(τ) = δj,i+ˆxΞ(τ), which corresponds to applying an electric field E = −(dΞ(τ)/dτ)ˆx in the xˆ direction. The perturbed action reads

∑ ∑ ∫ ∑ ∑ ′ − ′ − ′ − ′ − ′ ′ − SΞ = M Tr ln[∂τ + µδ(τ, τ ) Σis(τ, τ )] M dτdτ Σis(τ τ )Gis(τ τ) i s= i s= ∫ ∑ ∑ [( ) ] 2 M ′ 1 ′ 2 ′ z + t dτdτ 1 − (A (τ) − A (τ )) δ ′ + i(A (τ) − A (τ ))σ ′ 2N 2 ij ij ss ij ij ss ⟨ij⟩ ss′=

′ ′ × Gjs′ (τ, τ )Gis(τ , τ) ∫ [ ] M ∑ ∑ 1 + t2 dτdτ ′ i(Ξ (τ) − Ξ (τ ′)) − (Ξ (τ) − Ξ (τ ′))2 G (τ, τ ′)G (τ ′, τ) 2N ij ij 2 ij ij js is ⟨ij⟩ s= ∫ ∑ ∑ 2 M ′ ′ ′ z ′ ′ − t dτdτ (A (τ) − A (τ ))(Ξ (τ) − Ξ (τ ))σ ′ G ′ (τ, τ )G (τ , τ), (7.46) 2N ij ij ij ij ss js is ⟨ij⟩ ss′=

0 where we integrated out the fermions and neglected the Ai as before. With the perturbed partition function

175 ∫ ′ − ′ D D D SΞ[A,G,Σ] ZΞ = A G Σ e , we then obtain the current-current correlator

1 δ2Z′ ⟨ ′ ⟩ Ξ Jx(τ)Jx(τ ) = ′ ′ ZΞ=0 δΞ(τ)δΞ(τ ) ∫ ( Ξ=0 ) −S′ [A,G,Σ] ′ ′ 2 ′ e Ξ=0 δS [A, G, Σ] δS [A, G, Σ] δ S [A, G, Σ] = DADGDΣ Ξ Ξ − Ξ . (7.47) Z′ δΞ(τ) δΞ(τ ′) δΞ(τ)δΞ(τ ′) Ξ=0 Ξ=0

The only term that survives after integrating out the fields (which makes G and Σ take their saddle-point values) is

[ ∫ ] M ⟨J (τ)J (τ ′)⟩ = −V t2 G(τ − τ ′)G(τ ′ − τ) − δ(τ − τ ′) dτ ′′G(τ − τ ′′)G(τ ′′ − τ) , (7.48) x x N where V is the system volume (number of sites in the hypercubic lattice). The right-hand-side of (7.48) automatically contains the sum of the paramagnetic and diamagnetic terms.

This gives rise to the DC conductivity, employing the scaling forms derived in Sec. 7.3.3,

( ) ⟨ ⟩ 2x DC − 1 JxJx (iΩm) ∼ M t σxx = lim , (7.49) V m→0 Ωm N T and the optical conductivity

( ) it 2x σ (Ω ≫ T ) = −2(M/N)C2 sin(πx)Γ(2x − 1) . (7.50) xx Ω

As discussed in Sec. 7.2, since the saddle-point value of G is gauge-independent at leading order in large-N, this answer for the conductivity is correctly gauge-invariant at leading order in large-N. Since the critical solution (7.35) is in general valid only for T ≪ t, the DC conductivity (7.49) is never parametrically in a bad-metallic regime of σDC ≪ 1 within the energy window of validity of the non-Fermi liquid solution.

176 7.6 Discussion

We have constructed a model of a disordered non-Fermi liquid phase of fermions at a finite density coupled to gapless fluctuating U(1) gauge fields, in a solvable large-N limit. In this non-Fermi liquid phase, both the fermion and photon Green’s functions are gapless, and decay as power-laws of time at long times. The power-law exponents are continuously tunable within a finite range, and, interestingly, depend upon the filling fraction of the fermions.

A special feature of our model is that the non-Fermi liquid phase arises under the combined effect of hopping and interaction terms, in contrast to the purely interacting SYK models. In the SYK models, the addition of quadratic hopping terms results in a weakly-interacting Fermi liquid solution in the infrared [42].

However, unlike the SYK models, in which the interaction between the fermions is instantaneous in the large-

N limit, the interaction between fermions in our model is retarded, mediated by gapless bosonic modes with singular propagators at low energies, leading to non-Fermi liquid behavior even in the presence of hopping terms [285].

Our model only possesses scale invariance in the infrared, and not the much more comprehensive time reparametrization invariance of the SYK models. At nonzero temperatures, this lack of time reparametrization symmetry in our model results in different finite temperature fermion Green’s functions from the conformal ones that appear in the generalized set of SYKq models with 1 < q/2 < 2-body interactions [41, 218].

Consequently, we do not expect our model to have as direct a holographic connection to AdS2 gravity as the

SYK models, or to display maximal chaos [36, 37, 218, 222, 225]. However, due to the quantum-critical scaling of the Green’s functions, we still expect the Lyapunov exponent for many-body quantum chaos to be an O(1) number times kBT/ℏ, similar to other models of fermions strongly coupled to fluctuating gauge fields [286].

The dynamic photon modes cause our model to have a much larger Hilbert space than the SYK models,

177 which only have fermions. This appears to allow for a finer spacing of the low-lying many-body energy levels than in the SYK models (which have a level spacing of ∼ e−N [39]), leading to parametrically larger values of entropy and specific heat at low temperatures, that are dominated by contributions from the photon modes.

We can view our model as a toy model of an ACL [29, 277], which is a candidate for the strange metal regime of the cuprate superconductors. This is an effective theory in which electrons are fractionalized into gapless fermionic chargons which carry their charge (but not spin), and gapped bosonic spinons that do not affect the low-energy fluctuations of the chargons. By defining our model on an N-dimensional hypercubic lattice, we obtain non-Fermi liquid charge transport properties, with a sub-linear power-law-in- temperature resistivity. The exponent of the power-law is continuously tunable as a function of the filling, and can approach linear-in-temperature for certain parameter ranges. This non-Fermi liquid has a ‘large

Fermi surface’, i.e. all M flavors of fermions are active and contribute to transport. This is in contrast to the SYK/Kondo-lattice models of non-Fermi liquids proposed in Refs. [265, 266], where only the itinerant fermions contribute to transport.

For future work, it would be interesting to see if some of the strategies employed here can be extended to construct solvable models of fermions at finite density and with quenched disorder interacting with gauge fields in 2+1 dimensions. Such models would of course be more realistic candidates for describing the phase diagram of the cuprates. It would also be interesting, if possible, to consider Higgs transitions out of ACLs in such models into weakly interacting ‘pseudogap’ phases with a reduced number of active fermions [29, 279], along the lines of the analysis in Appendix G.1.

178 It’s a cruel and random world, but the chaos is all so beautiful.

Hiromu Arakawa

8 Quantum chaos on a critical Fermi surface

8.1 Introduction

States of quantum matter without quasiparticle excitations are expected [53] to have a shortest-possible local thermalization or phase coherence time of order ℏ/kBT as T → 0, where T is the absolute temperature.

Much recent attention has recently focused on the related and more precise notion of a Lyapunov time, τL, the time to many-body quantum chaos [45]. By analogy with classical chaos, τL is a measure of the time over which the wavefunction of a quantum system is scrambled by an initial perturbation. This scrambling can be measured by considering the magnitude-squared of the commutator of two observables a time t apart

[45, 46]: the growth of the commutator with t is then a measure of how the quantum state at the later

179 time has been perturbed since the initial observation. In chaotic systems, and with a suitable choice of observables, the growth is initially exponential, ∼ exp(t/τL), and this defines τL. With some reasonable physical assumptions on states without quasiparticles, it has been established that this time obeys a lower bound [46] ℏ τL ≥ ; (8.1) 2πkBT

(henceforth, we set kB = ℏ = 1). The lower bound is saturated in quantum matter states holographically dual to Einstein gravity [287], and in the SYK model of a strange metal [35, 36, 40, 218]. Relativistic theories in a vector large-N limit provide a weakly-coupled realization of states without quasiparticles, and in these cases τL ∼ N/T [50, 102, 103], which is larger than the bound in Eq. (8.1) but still of order 1/T . Fermi

2 liquids have quasiparticles, and their τL ∼ 1/T is parameterically larger than Eq. (8.1) as T → 0 [51, 288].

In general we expect that τL is of order 1/T only in sytems without quasiparticle excitations.

In this chapter, we turn our attention to non-Fermi liquid states of widespread interest in condensed matter physics. The canonical example we shall examine is that of N species of fermions at a non-zero density coupled to a U(1) gauge field in two spatial dimensions. Such a theory has a Fermi surface in momentum space which survives in the presence of the gauge field*, even though the fermionic quasiparticles do not.

Closely related theories apply to a wide class of problems with a critical Fermi surface, and we expect that our results can be extended to these cases too.

It has been recognized for some time [76] that the naive vector 1/N expansion of the critical Fermi surface problem breaks down at higher-loop orders (beyond three loops in the fermion self energy). This is in strong contrast to the behavior of relativistic theories at zero density in which the vector 1/N expansion is well behaved. This indicates the large N theory of a critical Fermi surface is strongly-coupled. Strong- coupling effects have been examined by carefully studying higher loops, or by alternative expansion methods

* The Fermi surface is defined by the locus of points where the inverse fermion Green’s function vanishes, and is typically computed in the gauge ∇ · a = 0: this yields the same Fermi surface as in the closely-related problem of a Fermi surface coupled to Ising-nematic order.

180 [28, 79, 289, 290], and in the end the results are similar to those in an random-phase approximation (RPA) theory [27, 31, 71]. So far, the main new effect discovered at strong coupling is a small fermion anomalous dimension, but this will not be important for our purposes here.

Here, we shall use an extended RPA theory to compute the Lyapunov time, and the associated butterfly velocity vB [54–57, 287, 291–293], for the critical Fermi surface in two spatial dimensions. As T → 0, we find for the Lyapunov rate λL ≡ 1/τL

λL ≈ 2.48 T (8.2)

which obeys the bound λL ≤ 2πT in Eq. (8.1). Notably the value of λL for the critical Fermi surface is independent of the gauge coupling constant, e, and also of N. This supports the conclusion [76] that this theory is strongly coupled in the N → ∞ limit. Our result for the butterfly velocity is more complicated; as

T → 0 NT 1/3 v5/3 v ≈ 4.10 F . (8.3) B e4/3 γ1/3

This depends upon both N and e, and also on the Fermi velocity, vF , and the Fermi surface curvature, γ.

Blake [54, 55] has recently suggested, using holographic examples, that there is a universal relation between transport properties, as characterized by the energy and charge diffusivities [261], and the parameters characterizing quantum chaos vB, and λL. For the critical Fermi surface being studied here, momentum is conserved by the critical theory, and so the electric conductivity is sensitive to additional perturbations which relax momentum [62, 293]. However, the thermal conductivity is well-defined and finite in the nonrelativistic critical theory [69, 161] even with momentum exactly conserved. So we may define a energy diffusivity, DE, which we compute building upon existing work [31, 248], and find

v2 DE ≈ 0.42 B . (8.4) λL

181 Notably, the factors of e, N, vF and γ in Eq. (8.3) cancel precisely in the relationship Eq. (8.4). This supports the universality of the relationship between thermal transport and quantum chaos in strongly-coupled states without quasiparticles.

A simple intuitive picture of this connection between chaos and transport follows from the recognizing that quantum chaos is intimately linked to the loss of phase coherence from electron-electron interactions. As the time derivative of the local phase is determined by the local energy, phase fluctuations and chaos are linked to interaction-induced energy fluctuations, and hence thermal transport. On the other hand, other physical ingredients enter into the transport of other conserved charges, and so we see no reason for a universal connection between chaos and charge transport.

8.2 Model

We consider a single patch of a Fermi surface with N fermion flavors, ψj, coupled to a U(1) gauge boson in two spatial dimensions: this is described by the “chiral non-Fermi liquid” model [81] (Fig. 8.1a). The

(Euclidean) action is given by

  ∫ 3 ∑N d k † N S =  ψ (k)(−iηk + ϵ )ψ (k) + ϕ(k)(c |k |/|k | + k2)ϕ(−k) e (2π)3 j 0 k j 2 b 0 y y j=1 ∫ ∫ 3 ∑N 3 d k d q † + e ϕ(q)ψ (k + q)ψ (k), (2π)3 (2π)3 j j j=1

2 2 ϵk = vF kx + γky , cb = e /(8πvF γ). (8.5)

This is derived from the action of a Fermi surface coupled to a U(1) gauge field with gauge coupling constant e. We only include the transverse gauge fluctuations in the gauge ∇ · a = 0, in which cause the gauge field reduces to a single boson ϕ representing the component of the gauge field perpendicular to the Fermi surface.

We have already included the one-loop boson self energy in Se. Unless explicitly mentioned, we shall set

182 the Fermi velocity vF and the Fermi surface curvature γ to unity in the rest of this chapter. These factors can be restored by appropriately tracing them through the computations. An advantage of this model is that the one-loop scaling structure of the boson and fermion Green’s functions is “exact”. As there is only a single patch, the one-loop scaling structure is not destroyed by the coupling of different patches at higher loop orders [28]. However, this theory is still not fully controllable via the large-N expansion: IR divergences in higher loop diagrams, such as the three-loop fermion self energy, enhance their coefficients by powers of

N. Ultimately, all planar diagrams must be taken into account [76]. A version of this model that combines two antipodal patches of the Fermi surface is amenable to a more controlled ϵ = 5/2 − d expansion [79].

However, our analysis cannot be performed easily with this dimensionally regularized construction, so we will restrict ourselves to the d = 2 RPA theory. Despite its flaws, the RPA theory has correctly determined other physical features of this theory, such as the scaling of the optical conductivity [27, 161] which agrees with the ϵ = 5/2 − d expansion [161].

Figure 8.1: (a) Fermi surface patch and coordinate system (b) The complex-time contour C used for evaluating out-of- time-order correlation functions. It contains forward and backward time evolution along two real time folds separated by iβ/2, and imaginary time evolution between the folds.

The bare frequency dependent term in the fermion propagator is irrelevant in the scaling limit and is hence multiplied by the positive infinitesimal η. However, the presence of this term might lead to crossovers in the

183 quantities that we compute at high temperatures. The above action is invariant under the rescaling

−1 −1/2 −3/2 kx → b kx, ky → b ky, k0 → b k0,

e → e, ψ → b2ψ, ϕ → b2ϕ. (8.6)

The coupling e is thus dimensionless, and the dynamical critical exponent is z = 3/2.

Since we will need to perform all computations at finite temperature, it is imperative that we understand what the finite-temperature Green’s functions are. In the above patch theory, the gauge boson does not acquire a thermal mass due to gauge invariance [62]. However, we will nevertheless add a very small “mass” by hand to use as a regulator. The boson Green’s function then is

|ky|/N D(k) = 3 2 . (8.7) |ky| + cb|k0| + m

This boson Green’s function may then be used to derive the thermally corrected fermion Green’s function via the one-loop self-energy starting from free fermions [75] (Appendix H.1)

2 1( ) √e T G(k) = | |− , µ(T ) = , (8.8) 2 − cf 2/3 k0 πT sgn(k0) − µ(T ) 3 3m2/3 kx + ky i N sgn(k0)T H1/3 2πT isgn(k0) N

√ 5/3 4/3 2 (cf = 2 e /(3 3)) where µ(T ) is generated by m cutting off an IR divergence coming from the zeroth boson Matsubara frequency, and H1/3(x) is the analytically continued harmonic number of order 1/3, with

∑n 1 H (n ∈ Z+) ≡ ,H (z) = ζ(r) − ζ(r, z + 1). (8.9) r jr r j=1

This thermally corrected Green’s function is not exact owing to the uncontrolled nature of the large-N expansion. Higher (three and beyond) loop corrections to the fermion self energy also contain terms that are

184 ultimately of order 1/N, which will modify the self energy but should leave the relative scalings of frequency, momentum and temperature unchanged [76]. The same is also true for various other diagrams. As such, the numerical prefactors in the Lyapunov exponent and butterfly velocity that we determine may not be exact, but we should be able to correctly deduce their scaling properties.

8.3 Scrambling and the Lyapunov exponent

To study out-of-time-order correlation functions, we define the path integral on a contour C which runs along both the real and imaginary time directions, with two real-time folds separated by iβ/2 [50, 218] (Fig. 8.1b).

The generating functional is given by

∫ ¯ Z = Dψ¯DψDϕeiS[ψ,ψ,ϕ]. (8.10) C

To measure scrambling, we will use fermionic operators, and hence we replace the commutators [45] by anti-commutators. We will evaluate the index-averaged squared anticommutator [50, 218]

∑N ∫ [ ] ∫ 1 † † f(t) = θ(t) d2x Tr e−βH/2{ψ (x, t), ψ (0)}e−βH/2{ψ (x, t), ψ (0)}† = d2x f(t, x). N 2 i j i j i,j=1 (8.11)

This function is real and invariant under local U(1) gauge transformations of the ψs. The staggered factors of e−βH/2 place two of the field operators on each of the real time folds. f(t) contains the out-of-time- ordered correlation function ⟨ψ(x, t)ψ†(0)ψ†(x, t)ψ(0)⟩ that describes scrambling. The anticommutators simplify the evaluation in comparison to the correlation function of just the four fermionic operators. f(t) then measures how the operators “spread” as a function of time. At t = 0, the anticommutators vanish for x ≠ 0. At later times, the operators become non-local under the time evolution, leading to a growth of the

185 function. It is conjectured [46] that at short times

f(t) ∼ eλLt + ... , (8.12)

where 0 ≤ λL ≤ 2πT is the Lyapunov exponent. Our goal is to compute λL. At long times, which we are not interested in, f(t) saturates to some finite asymptotic value. Formally, to precisely define λL, we need the growing exponential in (8.12) to have a small prefactor. This can be provided here by examining spatially separated correlators (which we shall do in Section 8.4), although not by the large N limit. Operationally, for now, we will compute f(t) by using diagrams similar to those employed in relativistic theories [50].

The approach described in Ref. [50] involves summing a series of diagrams to obtain f(t). The simplest subset of these is a ladder series (Fig. 8.2), with the “rails” of the ladder defined on the real-time folds, and the “rungs” connecting times separated by iβ/2. The interaction vertices are integrated only over the real- time folds as an approximation to minimize technical complexity; more general placements are expected to make corrections to the thermal state that should not affect λL. The end result is that one uses retarded

Green’s functions for the rails (since the real time folds involve both forward and backward evolution) and

Wightman functions for the rungs [50]:

[ ] GR(x, t) = −iθ(t)Tr e−βH {ψ(x, t), ψ†(0)} ,

1 GR(k) = ( ) , c 2 − f 2/3 − ik0+πT − µ(T ) kx + ky i N T H1/3 2πT i N [ ] [ ] GW (x, t) = Tr e−βH ψ(x, t)ψ†(0, iβ/2) = Tr e−βH/2ψ(x, t)e−βH/2ψ†(0) ,

A(k) GW (k) = , βk0 2 cosh 2

186 Figure 8.2: The Bethe-Salpeter equation for f(ω) at leading naive order in 1/N. Solid lines are fermion propagators and dashed lines are boson propagators. The arrows indicate the directions of momentum flow used in the equations in the text. For the fermion lines, advanced Green’s functions are used for the upper rails and retarded ones for the lower rails, as can be seen from Eq. (8.11). The third diagram on the right hand side is the same order in 1/N as the second despite having two boson propagators, because it involves summing over the flavors j.

( ) DR(x, t) = −iθ(t)Tr e−βH [ϕ(x, t), ϕ(0)] , | | R ky /N D (k) = 3 2 , |ky| − icbk0 + m [ ] [ ] DW (x, t) = Tr e−βH ϕ(x, t)ϕ(0, iβ/2) = Tr e−βH/2ϕ(x, t)e−βH/2ϕ(0) ,

B(k) 1 c k |k | DW (k) = = b 0 y . (8.13) βk0 N βk0 (|k |3 + m2)2 + c2k2 2 sinh 2 sinh 2 y b 0

(A is the fermion spectral function and B is the boson spectral function). For an explicit derivation of the

Wightman functions see Appendix H.2. There are two types of rungs at leading order in 1/N: one is simply the boson Wightman function. The other is a “box” that contains fermion Wightman functions and retarded boson functions.

The first diagram in the ladder series which has no rungs is given by

∫ 1 f (t) = d2x |GR(x, t)|2, 0 N ∫ 1 d3k f (ω) = GR(k)GR∗(k − ω) 0 N (2π)3 ∫ i dk dk 1 = y 0 [ ( ) ( )] . (8.14) 2 c − N (2π) f 2/3 − ik0+πT − i(ω k0)+πT µ(T ) i N T H1/3 2πT + H1/3 2πT + 2i N

This bare term remarkably ends up being O(1). Since m is tiny, µ(T ) → +∞. In the time domain, this thus describes a function that decays exponentially very quickly.

187 We have the Bethe-Salpeter equation of the ladder series

∫ 1 d3k f(ω) = f(ω, k) N (2π)3 ∫ [ ∫ ] 1 d3k d3k′ ( ) = GR(k)GR∗(k − ω) 1 + e2DW (k − k′) + K (k, k′, ω) f(ω, k′) , N (2π)3 (2π)3 2 [ ∫ ] d3k′ ( ) f(ω, k) = GR(k)GR∗(k − ω) 1 + e2DW (k − k′) + K (k, k′, ω) f(ω, k′) , (8.15) (2π)3 2

The sign of the e2DW term is +1, coming from a factor of −i2 arising from the i’s in in the interaction vertex of S. We need to solve this integral equation to determine the behavior of f(t). We note that as in Ref. [50], the condition for f(t) to grow exponentially is that the ladder sum be invariant under the addition of an extra unit to the ladder, i.e.

∫ d3k′ ( ) f(ω, k) = GR(k)GR∗(k − ω) e2DW (k − k′) + K (k, k′, ω) f(ω, k′), (8.16) (2π)3 2

We have

∫ d3k K (k, k′, ω) = Ne4 1 DR(k )DR∗(k − ω)GW (k − k )GW (k′ − k ). (8.17) 2 (2π)3 1 1 0 1 0 1

The overall sign of this contribution is also i2(−i)2 = 1, where the factors of i again come from the four interaction vertices. Here we use the bare fermion Wightman functions, as the self energy corrections will come in at higher orders in 1/N. As the integral is free of IR divergences, the overall power of 1/N in this contribution is not enhanced and this simplification should be safe. In the bare fermion Wightman functions, we drop the frequency dependent term that is irrelevant at low frequencies by sending η → 0, to obtain the quantum critical scaling limit,

2 A(k) πδ(kx + k ) GW (k) = → y . (8.18) 0 βk0 βk0 2 cosh 2 cosh 2

188 (A is the fermion spectral function). There is a cosh instead of a sinh in the fermion Wightman function

(Appendix H.2). We then have

∫ 4 3 2 ′ e d k1 k1y K2(k, k , ω) = 3 3 2 3 2 N (2π) (|k1y| − icbk10 + m )(|k1y| + icb(k10 − ω) + m ) π2δ(k − k + (k − k )2)δ(k′ − k + (k′ − k )2) × x 1x y 1y x 1x y 1y ′ − . (8.19) k0−k10 k0 k10 cosh 2T cosh 2T

2 Since there are no IR divergences, we drop the m s. Doing the k1x integral followed by the k1y one, this simplifies to

∫ ′ 4 − ′ 2| − |3 ′ e (ϵk ϵk ) ky ky K2(k, k , ω) = dk10 | − ′ |3 − | − ′ |3 | − ′ |3 − | − ′ |3 πN ( ϵk ϵk 8icbk10 ky ky )( ϵk ϵk + 8icb(k10 ω) ky ky )

× 1 ′ − . (8.20) k0−k10 k0 k10 cosh 2T cosh 2T

Due to the sliding symmetry along the Fermi surface [28], we expect the eigenfunction that we are interested in to obey f(ω, k) = f(ω, k0, ϵk). This can be proven by induction considering the series of diagrams that

′ → ′ − ′2 ′ → ′ ′ we sum. We can then shift kx kx ky followed by ky ky + ky and integrate over ky

∫ ∫ ∫ ( ) 3 ′ 4 ′ ′ 1/3 1/3 d k e dk dk k10 (−ik10) − (i(k10 − ω)) K (k, k′, ω)f(ω, k′) = √ 0 x dk 3 2 4/3 2 10 4/3 1/3 (2π) (2π) (−ik10) (i(k10 − ω)) (2k10 − ω) 24π 3cb N ′ ′ × f(ω, k0, kx) ′ − , (8.21) k0−k10 k0 k10 cosh 2T cosh 2T

189 and

∫ 3 ′ 2 d k W − ′ ′ e 3 D (k k )f(ω, k ) (2π)   c (k′ −k ) [ ′ ′ b 0 0 − ′ ∫ f(ω, k , k ) ′ − 2cbT f(ω, k0, k ) 2 3 ′ 0 x β(k k0) x e d k  sinh 0  | ′ − | 2  = lim  ky ky ′  m→0 3 | ′ − |3 2 2 2 − 2 N (2π) ( ky ky + m ) + cb (k0 k0) ∫ ] 2µ(T ) dk′ + x f(ω, k , k′ ) , (8.22) e2 2π 0 x where we added and subtracted terms to make the IR divergences as m → 0 explicit. If we expand the

′ → numerator of the integrand in round brackets in the above equation for k0 k0, we see that the integral is finite and free of IR divergences.

′ Interestingly, both pieces of the kernel no longer depend on kx and kx. Thus we can integrate both sides ∫ ′ ≡ dkx of the equation over kx and kx to get an equation for f(ω, k0) 2π f(ω, k0, kx). From Eq. (8.14), we can see that the IR divergent piece ∝ µ(T ) cancels out. The dependence on N also cancels out. We finally get

′ − ′ cb(k0 k0) ∫ ′ − ′ ′ f(ω, k0) β(k −k ) 2cbT f(ω, k0) dk dk sinh 0 0 2 0 y | ′ | 2 e lim ky ′ m→0 2 | ′ |3 2 2 2 − 2 (2π) ( ky + m ) + cb (k0 k0) ∫ ( ) 4 ′ − 1/3 − − 1/3 ′ e dk0dk10 k10 ( ik10) (i(k10 ω)) f(ω, k0) + √ ′ 4/3 − 4/3 − 1/3 − k −k k −k10 24π 3c 2π ( ik10) (i(k10 ω)) (2k10 ω) cosh 0 10 cosh 0 (b ( ) ( )) 2T 2T ik + πT i(ω − k ) + πT = c T 2/3 H − 0 + H − 0 f(ω, k ). (8.23) f 1/3 2πT 1/3 2πT 0

As a matrix equation, this is of the form M(ω)f(ω) = 0. Since we are looking for a positive growth exponent, we need to numerically find solutions of this equation on the positive imaginary ω axis. The analytic continuations of the self-energies that we made are still valid as long as Im[ω] > 0. The largest solution will provide the growth exponent λL. We can see from the above equation and from the quantum

′ ∼ ′ ∼ 2/3 1/3 ∝ critical scaling k0, k0 T , ky, ky e T that λL T and is independent of e. The numerical solution

190 to this equation is detailed in Appendix H.4. We find that λL ≈ 2.48 T , which is well within the bound of Ref. [46]. We further see that λL is not suppressed by powers of 1/N, unlike other vector models in the large-N limit. This indicates that this theory is strongly coupled at the lowest energy scales, even for large values of N.

The cancellation of the IR divergent piece between the self-energy and ladder diagrams has an important physical meaning. Besides being required by gauge invariance (as f(t, x) is gauge invariant), it indicates that “classical” processes do not contribute to many-body quantum chaos: The IR divergent terms arise from classical collisions in which the frequency of the boson is zero. In this limit, the boson behaves like a thermally (but not quantum-mechanically) fluctuating random potential for the fermions, each instance of which can be described by an integrable quadratic Hamiltonian, and is hence unable to induce chaos.

At high temperatures, when NT 1/3/e4/3 ≫ 1, we may no longer be able to neglect the bare frequency dependent term in the fermion propagators. This would essentially amount to adding a term ∼ Nωf(ω, k0)

4/3 2/3 to the right hand side of Eq. (8.23). Counting powers, we then might expect λL ∼ e T /N. In Ap- pendix H.3 we consider a few higher order (in 1/N) corrections to the ladder series and show that some of them are insignificant.

8.4 The butterfly effect and energy diffusion

8.4.1 Butterfly velocity

The out-of-time-order correlation function evaluated at spatially separated points characterizes the divergence of phase space trajectories in both space and time. This process is described by the function f(t, x) defined in Eq. (8.11), which is the same as the function f(t) we used to determine λL except for the integration over spatial coordinates. This function should contain a traveling wave term that propagates with a speed known as the “butterfly velocity” [56]. In order to compute this function we will need to evaluate

191 the ladder diagrams at a finite external momentum p. For simplicity, and since the component of the Fermi velocity perpendicular to the Fermi surface dominates the one parallel to the Fermi surface, we will take the external momentum to also be perpendicular to the Fermi surface. This will allow us to determine the component of the butterfly velocity perpendicular to the Fermi surface (vB⊥).

Repeating the same steps that led to the derivation of Eq. (8.23), we simply obtain

′ − ′ cb(k0 k0) ∫ ′ − ′ ′ f(px, ω, k0) β(k −k ) 2cbT f(px, ω, k0) dk dk sinh 0 0 2 0 y | ′ | 2 e lim ky ′ m→0 2 | ′ |3 2 2 2 − 2 (2π) ( ky + m ) + cb (k0 k0) ∫ ( ) 4 ′ − 1/3 − − 1/3 ′ e dk0dk10 k10 ( ik10) (i(k10 ω)) f(px, ω, k0) + √ ′ 4/3 − 4/3 − 1/3 − k −k k −k10 24π 3c 2π ( ik10) (i(k10 ω)) (2k10 ω) cosh 0 10 cosh 0 (b ( ) ( )) 2T 2T ik + πT i(ω − k ) + πT = c T 2/3 H − 0 + H − 0 f(p , ω, k ) + iNp f(p , ω, k ). f 1/3 2πT 1/3 2πT x 0 x x 0

(8.24)

4/3 2/3 For small px, we expect the change in exponent δλL/T ∼ −iNpx/(e T ). This implies that

∫ ∫ 1/3 λ t dpx ip (x−v ⊥t) NT dy f(t, x) ∼ e L g(t, Np )e x B , v ⊥ ∼ . (8.25) 2π x B e4/3

The structure of the above equation indicates that chaos propagates as a wave pulse that travels at the butterfly velocity. The wave pulse is not a soliton and broadens as it moves [56]; this is encoded in the function

1−1/z g(t, Npx) and further details are provided in Appendix H.4. Note that this shows vB⊥ ∼ T , which can also be straightforwardly derived by using the appropriate scalings of space and time, [x] = −1 and [t] = −z,

4/3 2/3 and is also seen in holographic models [54]. Numerically we find that δλL/T ≈ −4.10(iNpx/(e T )), giving the result of Eq. (8.3) once the factors of Fermi velocity vF and Fermi surface curvature γ are restored.

(Appendix H.4).

This is again strictly valid only at the lowest temperatures, where NT 1/3/e4/3 ≪ 1. Thus the butterfly velocity cannot be arbitrarily large in the large-N limit. When NT 1/3/e4/3 ∼ 1, the structure of the fermion

192 propagator indicates that there may be a crossover to a z = 1 regime, in which vB⊥ will become a constant independent of N and T .

With the scalings [y] = −1/2 and [t] = −z, we see that the component parallel to the Fermi surface,

2/3 vB∥ ∼ T , which is smaller than vB⊥ at low temperatures. Then the butterfly effect will be dominated by propagation perpendicular to the Fermi surface in the scaling limit.

8.4.2 Energy diffusion

It has been conjectured, and shown in holographic models [54, 261] that the butterfly effect controls diffusive transport. The thermal diffusivity κ v2 DE = ∼ B , (8.26) CV 2πT

where κ is the thermal conductivity and CV is the specific heat at fixed density. In holographic theories

E ∼ 2 λL = 2πT , so a more appropriate phrasing of the above equation is D vB/λL [294]. We can compute

CV using the free energy of the fermions (the contribution of the boson is expected to be subleading at low temperatures [161]) ∫ ∑ d2k ∂2F F = −NT ln G˜−1(k),C = −T , (8.27) (2π)2 V ∂T 2 k0 where we use the one-loop dressed fermion propagator at zero temperature [28],

c˜ 3c G˜−1(k) = k + k2 − i f sgn(k )|k |2/3, c˜ = f . (8.28) x y N 0 0 f 2(2π)2/3

This computation is carried out in Appendix H.5. We obtain

∫ 10(22/3 − 1) γ1/3 dk C = Γ(5/3)ζ(5/3)T 2/3e4/3 y , (8.29) V 9(2π)2/3 2/3 2π vF

where we have again restored vF and γ.

193 Since the theory of a single Fermi surface patch is chiral, currents are non-zero even in equilibrium. We must thus define conductivities with respect to the additional change in these currents when electric fields and temperature gradients are applied. The thermal conductivity κ is finite in the DC limit as it is defined under conditions where no additional electrical current flows. This can be achieved by simultaneously applying an electric field and a temperature gradient such that there is only an additional energy current but no additional electrical current. We have α2T κ =κ ¯ − , (8.30) σ where α, κ¯ are the thermoelectric conductivities and σ is the electrical conductivity. Any infinities in the DC limit arising from momentum conservation cancel between κ¯ and the other term, yielding a finite κ [161]. κ¯ may be obtained from the Kubo formula [295]

[ ] ∂ E E + κ¯⊥ = −βRe lim i⟨J⊥ J⊥ ⟩(iq0 → ω + i0 ) , (8.31) ω→0 ∂ω with the energy current

∫ 3 ( ) E − d k q0 ∂ϵk † J⊥ (iq0) = i 3 k0 + ψ (k + q0)ψ(k) (2π) 2 ∂kx ∫ ( ) d3k q = −i k + 0 ψ†(k + q )ψ(k). (8.32) (2π)3 0 2 0

We compute the conductivities using the one-loop dressed fermion propagators in Appendix H.5 (The boson again does not contribute directly due to the absence of an x-dependent term in its dispersion). The simplest vertex correction vanishes due to the structure of the fermion dispersions and other corrections are in general suppressed by powers of N. Since we effectively have particle-hole symmetry about the Fermi surface in this system with an infinitely deep Fermi sea and infinitely large bandwidth, the thermoelectric coefficient

194 E α⊥ ∝ ⟨J⊥ J⊥⟩ (where J⊥ is the charge current) vanishes, so κ⊥ =κ ¯⊥. We obtain (restoring vF and γ)

∫ 2 1/3 8/3 N T vF dky κ⊥ ≈ 0.28 . 1/3 (8.33) c˜f γ 2π

Using Eqs. (8.2), (8.3), (8.26), (8.29) we then see that

N 2 v10/3 v2 DE ≈ 2.83 F ≈ 0.42 B⊥ . ⊥ 8/3 1/3 2/3 (8.34) e T γ λL

The factors of powers of T,N and e match exactly on both sides of the equation and the constant of pro-

E 2 O portionality between D⊥ and vB⊥/λL is an (1) number. This strongly indicates that the butterfly effect is responsible for diffusive energy transport in this theory. The DC electrical conductivity is however infinite due to translational invariance, and hence, unfortunately, such a statement cannot be made for charge ∫ dky transport in this model. Note that the hyperscaling violating factor 2π [161] cancels between κ⊥ and CV .

E E However, if we consider κ∥, this does not happen due to the additional ky dependence in J∥ . Thus D∥ will

2 not be given by vB∥/λL.

8.5 Discussion

We have computed the Lyapunov exponent λL and butterfly velocity vB for a single patch of a Fermi surface with N fermion flavors coupled to a U(1) gauge field. At the lowest energy scales, this theory is strongly coupled regardless of the value of N, and we hence find that λL is independent of N to leading order in

1/N. The proposed universal bound of λL ≤ 2πT is also obeyed. While the 1/N expansion is not fully controllable, it has nevertheless been capable of correctly determining many physical features of this theory in the past. We find that the butterfly velocity is dominated by propagation perpendicular to the Fermi surface,

1/3 and that vB⊥ ∼ NT . Most interestingly, we find that the butterfly effect controls diffusive transport in

195 E ∝ 2 this model, with the thermal diffusivity D⊥ vB⊥/λL. Our results are valid at the lowest energy scales, at which the quantum critical scaling holds. At high temperatures, we might expect λL to cross over to a

2/3 slower T /N scaling, and that vB⊥ simply becomes a constant independent of N and T . While technically much more complex to obtain, it would be interesting to compare the results derived from a more controlled calculation, such as the ϵ = 5/2 − d expansion for the two-patch version of the problem, with our results.

Finally, we note recent experimental measurements of thermal diffusivity in the cuprates [18] which find a strong coupling to phonons. It would be of interest to extend the chaos theories to include the electron- phonon coupling.

196 The caterpillar does all the work, but the butterfly gets all the publicity.

George Carlin

9 Quantum butterfly effect in weakly interacting

diffusive metals

9.1 Introduction

Elucidating the physics of thermalization in isolated quantum systems [220, 221, 296, 297] represents an ongoing challenge in quantum many-body physics, and great progress has been made in recent years due to advances in both theory and experiments [298–305]. In this chapter we are interested in the process of thermalization in interacting disordered metals, specifically in the physics of quantum information scrambling. Starting from a local perturbation, scrambling describes the spreading of quantum entanglement and

197 information across all of the degrees of freedom in a system [306–309], leading to a loss of memory of the initial state. The onset of scrambling is associated with the growth of chaos and is an intermediate step in the eventual global thermalization at late times of an isolated quantum many-body system.

It has become clear recently that certain special correlation functions can probe the onset of scrambling

[287, 310]. While such correlators first appeared in the literature many decades ago [45], there has been a revival in their interest, partly due to their relevance in studying information scrambling in black holes

[57, 287, 310]. For two local operators X and Y in a system described by a Hamiltonian H, these correlation functions are defined as

[ ] f(t) = Tr ρ [X(t),Y ]† [X(t),Y ] , (9.1)

where ρ ∝ e−H/T is the density matrix of an equilibrium state at temperature T and X(t) = eiHtXe−iHt.

The intuition for considering this object is that local operators must grow in time if information is to spread across a system and the commutator measures this growth. Furthermore, in order to access generic matrix elements of the commutator, one considers the average of the square of the commutator, f(t), which is non- negative and avoids phase cancellations. In contrast, the average of the commutator is a response function, and these tend to decay to zero at late times in a chaotic system.

A few comments about f(t) may be helpful. When expanding out f(t) in terms of 4-point functions one finds that it contains both time-ordered and out-of-time-order (OTO) pieces. When dealing with fermionic operators, it is more convenient to study instead the squared anti-commutator. For non-interacting fermions the anti-commutator is proportional to the single particle propagator and encodes causality. More generally, one can relate commutators of composite bosonic operators, e.g. fermion bilinears, to the basic fermion anticommutator. For a field-theory defined in the continuum, we use the ‘regulated’ version of the correlator above, where two of the operators have been moved halfway along the thermal circle to deal with spurious

198 divergences.

In a chaotic system with a local Hamiltonian, one expects f(t) to start out small when X and Y are spatially separated, and to grow exponentially in time, f(t) ∼ ϵ eλLt, where ϵ is a small parameter that may depend on time and the distance between X and Y . By considering an appropriate analytic continuation of f(t), one can show that there is a fundamental upper bound on λL(≤ 2πkBT/ℏ) [46]; black-holes and certain random fermion models [35, 310] saturate the bound. On the other hand in glassy systems, or in systems that simply fail to thermalize (but are not fully integrable), f(t) may have a power-law form [294].

While a measurement of such correlation functions is highly non-trivial, naively requiring a ‘time-machine’ in the laboratory, a few novel protocols have been proposed [311–313] and three preliminary experiments

[314–316] have already been carried out within the last couple of months.

In this chapter, we study scrambling in (weakly) interacting diffusive metals [317]. We consider the case of Coulomb interactions as well as short-range interactions in two and three spatial dimensions. Based on the general intuition that disorder slows the spread of charge and heat, one might also expect that operators spread more slowly in space in a disordered metal relative to a clean metal. Relatedly, we expect the effects of interactions to be enhanced relative to the clean metal since diffusive electrons move slowly compared to ballistic electrons and the effects of interactions can build up. We compute the growth exponent to lowest order in the strength of the interaction while carrying out an infinite resummation over disorder and indeed find that λL is larger at low T than the corresponding result for a clean Fermi liquid. We also find that chaos grows in a ballistic fashion, with a velocity that is parametrically smaller than the Fermi-velocity at low temperatures.

These computations confirm a recent argument [294] that even though the transport of charge and energy is diffusive in such metals, generic operators grow ballistically (see also Refs. [318–320] for a related observation in one-dimensional systems). This is not too surprising, since there is no reason for the motion of charge and energy to be tied to the growth of chaos in interacting systems; an extreme example being that

199 Figure 9.1: (a) Cartoon showing a snapshot at time t of the spread of chaos in an interacting diffusive metal. The fuzzy circles of radius ∝ (Dt)1/2 represent electrons diffusing through a background of impurities (small black dots). We make an analogy to the spread of an epidemic: An ‘infected’ electron inserted into the center of the figure at t = 0 diffuses outwards (fuzzy red circle). As it encounters other diffusing electrons, it infects them. These newly infected electrons further infect other electrons and so on (fuzzy green circles). The flight paths of the butterflies track the spread of the infection. The radius of the region containing infected electrons (bounded by the dashed red circle) grows ballistically as vBt. Although not shown in the figure, the electrons also have a finite lifespan, given by the inverse of the quasiparticle decay rate. This needs to be taken into account when considering the population of infected electrons as a function of time. The function f(t, x) is roughly equivalent to the local fraction of infected electrons at a point x. (b) The behavior of f(t, x) for one operator placed at the center of the figure (red dot) and the other at a position x shown as a function of x at a given time t. f(t, x) displays a light-cone (a time slice of which is bounded by the dashed red circle; this region exclusively contains infected particles) within which it has saturated and no longer grows. The radius of this region grows as vBt.

of a many-body localized (MBL) phase [321, 322] where partial scrambling occurs even in the absence of

any transport of charge and heat [294, 323–327]. When considering long range Coulomb interactions, it is

particularly interesting that we find ballistic growth of operators since the microscopic model does not have

a Lieb-Robinson bound [328]. There are variants of the Lieb-Robinson bound for systems with power law

interactions [329, 330], but these bounds allow exponential growth of operators with time while we find only

linear growth. √ 2 −1 On general scaling grounds, the butterfly velocity can be estimated to be vB ∼ Dγin, where D ∼ l τ

−1 is the diffusion constant (l ≡ mean-free path, τ ≈ elastic scattering rate) and γin is a small interaction-

200 induced inelastic scattering rate [294]. In the presence of weak interactions, we expect

− 2 f(t, x) ∼ eλLte x /(4Dt); (9.2)

the exponential growth reflects the onset of chaos in an ergodic system as discussed above while the latter contribution is a result of diffusion. Solving for f(t, R(t)) ∼ 1, where R(t) is a typical ‘operator-radius’— which, given an initial perturbation, defines the region in space over which information has spread over time

2 2 t—leads to R ∼ 4DλLt (Figure 9.1). One therefore obtains a light-cone like growth of f with a butterfly velocity √ vB = 4DλL. (9.3)

We show in this chapter, by carrying out a perturbative ‘ladder’ computation [50], that the disordered metal does obey Eq. (9.2), and the growth exponent, λL is indeed mostly given by the inelastic scattering rate with a singular temperature dependence. Note that the unitarity of quantum mechanics prevents f(t, x) from growing to values ≫ 1 and thus it saturates at very long times. Eq. (9.2) and the ladder computation are valid only for the pre-saturation growth of f.

The rest of this chapter is organized as follows: in Section 9.2, we define our model of interacting electrons in the presence of static disorder and set up the basic elements required for carrying out perturbation theory to leading order in the coupling strength. Section 9.3 deals with the perturbative computation of the important terms contributing to λL and vB for the case of Coulomb interactions in three spatial dimensions. In Section

9.4, we consider some additional effects in perturbation theory, as well as the case of short-range interactions, and show that our main results are unchanged by these modifications. Finally, in Section 9.5, we study the two-dimensional version of the problem, and point out a subtle difference between λL and the inelastic scattering rate. Unless explicitly mentioned, ℏ = kB = 1 in the rest of this chapter.

201 9.2 Preliminaries

We consider a model of N species of electrons in d ≥ 2 spatial dimensions subject to random potential disorder and weak interactions. We do not take any kind of large-N limit; N is a finite number (N = 2 for the case of spinful electrons). For most of this chapter, we shall focus on the physically relevant case of long- range Coulomb interactions in a metal; we also analyze the case of short-range interactions in Section 9.4.

From now on, we focus on the three-dimensional problem with d = 3 unless otherwise stated, but will analyze the case of two spatial dimensions with d = 2 in Section 9.5.

The Hamiltonian of interest is,

H = H0 + Hint, ∫ ( ) ∑N 2 † ∇ H = ddx ψ (x) U(x) − − µ ψ (x), 0 i 2m i i=1 ∑N ∫ d d ′ | − ′| † † ′ ′ Hint = d x d x Vb( x x )ψi (x)ψi(x)ψj (x )ψj(x ), (9.4) i,j=1

† where ψi (x) (ψi(x)) represent fermionic creation (annihilation) operators satisfying the usual anticommuta- tion algebra, µ is the chemical potential and m is the effective mass of the electrons. The disorder potential

U(x) breaks translational invariance and we assume

≪ ′ ≫ 2 d − ′ U(x)U(x ) = U0 δ (x x ), (9.5)

where ≪ ... ≫ denotes averaging over disorder realizations and U0 denotes the strength of disorder. We

′ shall treat the interaction, Vb(|x−x |), perturbatively, but will allow for strong disorder via the resummation of various classes of Feynman-diagrams with disorder lines. For Coulomb interactions in any number of

′ 2 ′ 2 dimensions Vb(|x − x |) = e /|x − x |, where e will be the small parameter in our perturbative treatment.

202 Let us now review the key features of the above theory before setting up the computation for the correlation functions describing chaos in Section 9.3. The remainder of this section closely follows the discussion in standard references (see e.g. Ref. [317]).

The bare electron imaginary time Green’s function after including the impurity self-energy (Figure 9.2a) is p2 i [G (ϵ , p)]−1 = −iϵ + − µ − sgn(ϵ ), (9.6) 0 n n 2m 2τ n

−1 2 where τ = U0 g(0) is the elastic electron scattering rate due to disorder (g(0) is the density of states at the ∫ d3p p2 − Fermi level; we use the convention g(0) = 2π (2π)3 δ( 2m µ)).

The real time Green’s functions are defined as (ψ(0) ≡ ψ(0, 0))

∫ d † d k dk · − θ(t)⟨{ψ (t, x), ψ (0)}⟩ = iδ GR(t, x) = iδ 0 GR(k , k)ei(k x k0t), i j ij ij (2π)d+1 0 ∫ d † † ∗ d k dk − · − θ(t)⟨{ψ (t, x), ψ (0)} ⟩ = −iδ GR (t, x) = −iδ 0 GA(k , k)e i(k x k0t). (9.7) i j ij ij (2π)d+1 0

As is well known in the theory of non-interacting disordered metals, the disorder averaged product of

Green’s functions in the particle-hole polarization bubble (density-density correlator) gives rise to the ‘dif-

−1 fuson’ mode at low frequencies and momenta (|ω|, vF q ≪ τ ),

dn Dq2 Π(ωm, q) = 2 , (9.8) dµ |ωm| + Dq

√ ∝ 2 2 with the non-interacting diffusion constant, D l /τ = vF τ (vF = 2µ/m is the Fermi velocity). The non-interacting compressibility is dn/dµ = Ng(0)/(2π). In the presence of interactions, the above diffuson mode introduces large vertex corrections (Figure 9.2b) to the electron-interaction vertices

2 −1 −1 Γ(q, ωm, ϵn) = (θ(ϵn(ϵn − ωm)) + θ(ϵn(ωm − ϵn))(|ωm| + Dq ) τ ), (9.9)

203 Figure 9.2: (a) The impurity self-energy leading to the elastic lifetime in Eq. (9.6) (b) Disorder correction to the electron interaction vertex in Eq. (9.9); here, and henceforth the electron lines contain the effect of the impurity self- energy (c) Dynamical screening of the interaction by the disorder-corrected polarization bubble in Eq. (9.10) (d) 2-in,2- out process that provides the inelastic electron lifetime; here, and henceforth the interaction line is the dynamically screened interaction.

where ωm = 2πmT and ϵn = π(2n + 1)T are Matsubara frequencies at a temperature T and effectively screens (Figure 9.2c) the long-range Coulomb interaction to,

4πe2 1 4πe2 |ω | + Dq2 V (ω , q) = = m . (9.10) m 2 4πe2 2 2 2 q q |ωm| + D(K + q ) 1 + Π(ωm, q) q2

In the above expression, K2 = 4πe2dn/dµ is proportional to the charge compressibility. Note that despite

2 2 2 the factor of e , we still treat K as an O(1) quantity while doing perturbation theory in e , since dn/dµ ∝ kF in d = 3 (kF ≡ Fermi-momentum) is large. Equivalently, this amounts to doing perturbation theory in

1/(dn/dµ) ∝ 1/(Ng(0)). Nevertheless we still have K ≪ kF .

Let us also review the computation of the disorder-averaged electron lifetime, which provides the inelastic scattering rate [317]. The process in Figure 9.2d, which includes the effects of the dynamically screened interaction and the vertex corrections, gives, via Fermi’s Golden Rule, the following expression for the out- relaxation rate or the ‘inelastic scattering rate’ γin(ϵ) for particles with energy ϵ of a given flavor i [247, 331,

204 332]:

∫ ∂n dϵ′dΩ d3q i,ϵ = −Ng(0) |V R(Ω, q)|2 ∂t 2π2 (2π)3 [out ] 1 2 × Re n n (ϵ′)(1 − n (ϵ − Ω)) × (1 − n (ϵ′ + Ω)) −iΩ + Dq2 i,ϵ F F F

≡ −ni,ϵγin(ϵ). (9.11)

Here nF (...) is the Fermi-Dirac distribution function. Here, the incoming particles are on-shell while the outgoing particles are allowed to be off-shell due to the dynamical interaction V R(Ω, q). At the Fermi level

(ϵ = 0), this simplifies to

∫ ∫ [ ] ∞ 3 2 Ng(0) dΩ Ω d q | R |2 1 γin(0) = 3 V (Ω, q) Re 2 2π −∞ π 2 sinh(βΩ) (2π) −iΩ + Dq ∫ ∫ 4 ∞ 3 ≈ 8πe Ng(0) dΩ Ω d q 1 4 3 2 2 4 K −∞ π 2 sinh(βΩ) (2π) Ω + D q √ (4 − 2)ζ(3/2)e2T 3/2 e2T 3/2 ≈ √ ≈ 0.674 , (9.12) 4 2πd3/2K2 D3/2K2 where we made the reasonable assumptions q ≪ K and Ω ∼ T ≪ DK2. We also used the non-interacting

2 result Ng(0) ≈ 2πdn/dµ, as corrections due to interactions will only correct γin(0) at higher orders in e .

d/2 At a finite energy away from the Fermi level, γin(ϵ) ∼ ϵ h(ϵ/T ) in d−spatial dimensions, where h(x) is a scaling function of x [317].

9.3 Many-body quantum chaos

To study the onset of quantum chaos for the model introduced in Eq. (9.4), we compute the flavor-averaged squared anticommutator of electron field operators perturbatively to leading non-trivial order in the coupling

205 e2,

∑N [ ] 1 † † f(t, x) = θ(t) Tr e−βH/2{ψ (t, x), ψ (0)}e−βH/2{ψ (t, x), ψ (0)}† . (9.13) N i j i j i,j=1

The prefactor of 1/N is inserted so that the bare contribution to f(t, x) is free of factors of N. The splitting of e−βH into two factors of e−βH/2 ensures that all operator insertions occur at distinct complex time points, thus avoiding short-distance divergences. The strict positivity of f(t, x) also guarantees exponential growth at a rate equal to that of the correlator where e−βH is not split [50, 52]. These “regularized” correlators have also been shown to obey fluctuation-dissipation-like relations [333]. Computing f(t, x) involves defining the action on a complex-time contour with real time folds separated by iβ/2 [50, 52, 286, 288]. We must then solve a Bethe-Salpeter equation arising from the resummation of different classes of ladder diagrams to determine f(ω, q), which after a Fourier transform yields information about the spatial and temporal structure of growth of chaos. An outline of the derivation of the Feynman rules for Eq. (9.13) required to set up the following diagrammatic calculation is presented in Appendix I.1.

Let us first quote the results for the non-interacting case, where V = 0. Here we do not expect chaotic growth of entanglement because the many-body state can be written as a Slater determinant of exact eigenstates of the one-body Hamiltonian. Summing the simplest class of ladder diagrams without any overlapping disorder rungs (Figure 9.3) yields the correct qualitative result, as shown in Appendix I.2. The final answer is

−x2/4Dt f(t, x) ∼ f0(t, x) + f1(t, x) e , (9.14)

−1 −3/2 where f0 is a rapidly decaying function of time with a rate set by τ and f1(t, x) ∼ (Dt) (in d = 3).

At times t ≫ τ, f(t, x) is dominated by the second term, which grows diffusively but then decays as a power law at long times. The diffusive behavior is expected as we have merely computed the particle-hole

206 Figure 9.3: (a) Resummation of disorder rungs. (b) Relation between L(ω, q) and f(ω, q). polarization bubble in real-time. As expected, there is no exponential growth.

We note here an important point, namely that we are actually computing ≪ f(t, x) ≫ averaged over different realizations of disorder. In a disordered metal for which the localization length of the eigenstates is far larger than the typical length scale over which the disordered potential varies, the disorder self-averages, and it hence makes sense to consider the disorder average of f(t, x) within a single copy of a system.

We now consider the effects of interactions, using a diagrammatic formalism which sums all the singular terms associated with diffuson and ‘Cooperon’ modes perturbatively in the intereaction strength [334–337].

Our perturbative computation sums all singular disorder corrections while working at O(e2) in the interaction, and is formally identical to the theory of Altshuler and Aronov [247]. We will examine two effects: (i) dissipative ‘self-energy’ corrections (Figure 9.4) that lead to decay, and, (ii) ‘ladder’ corrections (Figure 9.5) that lead to an exponential growth of the squared anticommutator [50]. In order to obtain a non-trivial chaotic growth, the effect of the latter has to overwhelm the former. Let us discuss them now one by one.

9.3.1 Self-energy corrections

The non-interacting L(ω, q) (Figure 9.3a) is given by

1 L(ω, q) = . (9.15) g(0)τ 2(−iω + Dq2)

207 Figure 9.4: The dominant Fock-type self-energy corrections to L(ω, q). Each diagram has a partner diagram generated by reflecting about the horizontal axis. Also not shown are the Hartree-type contributions, which are suppressed for sufficiently long-range interactions.

The dissipative self-energy corrections to the above quantity were considered by Castellani et. al. [334, 335].

These renormalize L(ω, q) at small ω, q to

1 Z L(ω, q) → . (9.16) g(0)τ 2 − ˜ 2 − R iω + Dq ΣL (0, 0)

For T ≠ 0, ΣL(0, 0) ≠ 0. The field renormalization Z and the renormalization of D → D˜ are not of particular concern to us as they will provide a correction to the growth exponent at O(e4); from now on we take Z = 1 and D˜ = D. The finite-temperature lifetime is important, and corrects the growth exponent downwards.

R To compute ΣL (0, 0) for the correlator spread across the two time folds, we note the Fock-type diagrams in

Figure 9.4 (and their partners obtained by reflection about the horizontal axis). We ignore the corresponding

2 2 Hartree-type diagrams, which are relatively suppressed by a factor of K /kF [247]. In the Fock diagrams, the time folds are connected only by static disorder lines and not the dynamical interaction, and hence there is no distinction between the two time fold correlator and the real-time retarded particle-hole correlator. Thus, in the end we only need to focus on the contribution arising from Figure 9.4(d) and its partner, as was noted in Refs. [334, 335], to get ΣL(0, 0). We have

∫ ∑ d3k V (Ω , k) m ΣL(ωl > 0, q) = 2T 3 2 . (9.17) (2π) D(k + q) + |Ωm + ωm| Ωm; ϵn<Ωm<ϵn+ωl ϵn<0

208 We do this sum by contour integration, noting the branch cut in V (Ωm, k) as iΩm crosses the real axis and that ϵn is a fermionic Matsubara frequency. The non-vanishing contribution upon analytically continuing

ϵn, ωl → 0 is [335]

∫ ∫ 3 ∞ R 2 d q dΩ 1 R ΣL (0, 0) = 2g(0)τ 3 L(Ω, q)Im[V (Ω, q)]. (9.18) (2π) −∞ π sinh βΩ

We have

4πe2 DK2Ω 4πe2 Ω Im[V R(Ω, q)] = − ≈ − . (9.19) q2 Ω2 + D2(K2 + q2)2 q2 DK2

Hence

∫ ∫ ∫ 2 ∞ ∞ 2 ∞ R ≈ − 4e 1 x − 2e 1 √ x ΣL (0, 0) 2 2 dk dx 2 = dx π DK −∞ sinh βx Dk − ix πD3/2K2 −∞ sinh βx −ix √ 0 (4 − 2)e2T 3/2ζ(3/2) T 3/2 = − √ ≈ −2.695 . (9.20) 2πD3/2K2 D3/2K2

− R Note that ΣL (0, 0) is also the decay rate of the Cooperon at zero external pair momentum and frequency [335,

338], which has been interpreted as the decay rate of electrons in exact eigenstates near the Fermi level [338–

340].

9.3.2 Ladder diagrams

In the ladder diagrams with interaction rungs (Figure 9.5), the disorder correction to the interaction vertices occurs on a single time fold. Therefore the second term of Eq. (9.9) does not apply, as it would correspond to the bare interaction vertex connecting Green’s functions on opposite time folds before being corrected by disorder, a possibility that is ruled out by the locality of the bare interaction vertex in time. Since the dynamic interaction (which can be interpreted to be mediated by a dynamically fluctuating boson) rung

209 connects two time folds on opposite sides of the thermal circle, its propagator is given by a bosonic Wightman function [50, 52, 286]

−2Im[V R(Ω, q)] 4πe2 DΩ K2 4πe2 Ω V W (Ω, q) = ( ) = ( ) ≈ ( ). βΩ q2 βΩ Ω2 + D2(K2 + q2)2 q2 2 βΩ 2 sinh 2 sinh 2 DK sinh 2

(9.21)

2 2 Note that only the dynamical part of the interaction V (ωm, q) − 4πe /q (which behaves like a Landau- damped boson) contributes to the Wightman function.

Direct Insertion.- We first consider the simplest summation of the ladder diagrams with alternating interaction and ‘diffuson’ rungs, L(ω, q), given by Figure 9.5a. By explicitly considering the series of diagrams, we see that the resulting unit, F , depends only upon the frequencies passing through it, but not the momenta.

The Bethe-Salpeter equation for F reads

′ − ′ F (ω, q, k0, k0) = L(ω, q)δ(k0 k0) + ∫ d3k d3k dk′′ L(ω, q) 1 2 0 V W (k − k′′, k − k )GR(k + ω, k + q)GA(k , k ) (2π)6 2π 0 0 1 2 0 0 1 0 0 1 × R ′′ A ′′ ′′ ′ G0 (k0 + ω, k2 + q)G0 (k0 , k2)F (ω, q, k0 , k0). (9.22)

The overall sign of the rung term is +1, coming from i2(−i)2, where the factors of i are generated by the

Hubbard-Stratonovich transformation of the Coulomb interaction to a fermion-boson interaction and the two real-time fermion-boson interaction vertices. After some manipulations, and assuming k1, k2 are close to

210 the Fermi surface, this becomes

1 1 F (ω, q, k , k′ ) ≈ δ(k − k′ ) + 0 0 g(0)τ 2 (−iω + Dq2) 0 0 ∫ ( ) 2 2 ′′ − ′′ 2 m 1 4πe dϵ1dϵ2 dk0 k0( k0 ) 16µ 2 2 2 4 − ′′ ln 2 g(0)τ (−iω + Dq ) DK (2π) 2π k0 k0 (ϵ1 − ϵ2) sinh 2T 1 1 × F (ω, q, k′′, k′ ), (9.23) − 2 1 − ′′ 2 1 0 0 (ϵ1 k0) + 4τ 2 (ϵ2 k0 ) + 4τ 2 where we also set ω, q = 0 in the internal Fermion Green’s functions, because the leading dependence on

ω, q for small ω, q comes from the 1/(−iω + Dq2) multiplying the integral. We can rewrite this for small

′′ ≪ −1 ≪ k0, k0 τ µ as

′ F (ω, q, k0, k0) ∫ 2 2 ′′ − ′′ ≈ 1 1 − ′ m ln(4µτ) 2e dk0 k0( k0 ) ′′ ′ 2 2 δ(k0 k0) + 2 2 − ′′ F (ω, q, k0 , k0). g(0)τ (−iω + Dq ) g(0) (−iω + Dq ) πDK 2π k0 k0 sinh 2T

(9.24)

As a matrix equation I/(g(0)τ 2) F = , (9.25) − 2 I − 2e2 ln(4µτ) A ( iω + Dq ) 2 0 vF DK where the elements of A0 are given by the integral kernel of the previous equation (9.24). Note that the translationally invariant structure of A0 implies plane wave eigenstates. The growing part of f(ω, q) is obtained by appending external lines to F , capping off the ladder sum and integrating over momenta (which ∫ d3k R A − ≪ −1 just provides two factors of g(0)τ = (2π)3 G0 (k0, k)G0 (k0 ω, k) for ω τ ) and frequencies (Fig- ∫ dk′ 2 dk0 0 ′ F A ure 9.5c): f(ω, q) = (g(0)τ) 2π 2π F (ω, q, k0, k0). Therefore and 0 have the same eigenvectors and

2 the largest positive eigenvalue of A0 (= πT ) provides the growth exponent

2 (0) ≈ 2πe 2 λL 2 T ln(4µτ). (9.26) vF DK

211 Figure 9.5: Ladder insertions at O(e2) which provide exponentially growing contributions to f(t, x). The ‘direct’ insertion in (a) provides a contribution that grows at a rate proportional to T 2, slower than the ‘exchange’ insertions in (b), which grow as T 3/2. The relationship between the function f(ω, q) and the ladder series is shown in (c).

Thus the growth exponent produced by the simplest ‘direct’ ladder insertion considered above is insufficient to overwhelm the T 3/2 decay rate from the self-energy corrections. We need to thus consider other ladder insertions at O(e2) and check to see if they generate an exponent that successfully competes with the decay rate. Henceforth, we ignore the contribution of A0 to the ladder sum.

Exchange insertion.- As discussed above, we need to consider additional ladder insertions at the same order in perturbation theory which at least compete with the previously computed decay rate. At O(e2),

212 these come from Figure 9.5b. The sum of the two insertions gives the following integral equation:

′ − ′ F (ω, q, k0, k0) = L(ω, q)δ(k0 k0)+ ∫ d3k d3k d3k dk′′ L(ω, q) 1 2 3 0 V W (k − k′′, k − k )L(k + ω − k′′, k + q − k ) (2π)3 (2π)3 (2π)3 2π 0 0 2 3 0 0 2 3

× R A R A ′′ − R ′′ G0 (k0 + ω, k1 + q)G (k0, k1)G0 (k0 + ω, k2 + q)G0 (k0 , k1 k2 + k3)G0 (k0 + ω, k3 + q)

× A ′′ ′′ ′ G0 (k0 , k3)F (ω, q, k0 , k0) ∫ d3k d3k d3k dk′′ + L(ω, q) 1 2 3 0 V W (k − k′′, k − k )L(k′′ + ω − k , k + q − k ) (2π)3 (2π)3 (2π)3 2π 0 0 3 2 0 0 2 3

× R A R A ′′ − R ′′ G0 (k0 + ω, k1 + q)G (k0, k1)G0 (k0 + ω, k2 + q)G0 (k0 , k1 k2 + k3)G0 (k0 + ω, k3 + q)

× A ′′ ′′ ′ G0 (k0 , k3)F (ω, q, k0 , k0). (9.27)

The overall sign of this contribution is +1 for the same reasons as above. Moreover, the two contributions are equal to each other. As before, we ignore the small ω, q contribution coming from within the integrand, and throw out the short-wavelength/high-frequency parts of the interaction. Since the interaction is long- ranged, the largest contribution to the integrals comes when the momentum k2 −k3 appearing in the internal interaction and in the ‘diffuson’ rungs is small compared to the momenta flowing through the internal fermion lines, which are O(kF ). We thus shift k3 → k3 +k2 and then ignore k3 everywhere except in the interaction and ‘diffuson’ rungs, which are singular at small k3. Then we have,

1 1 F (ω, q, k , k′ ) ≈ δ(k − k′ )+ 0 0 g(0)τ 2 (−iω + Dq2) 0 0 ∫ 2 3 ′′ − ′′ 8πe d k3 dϵ1dϵ2 dk0 k0( k0 ) 1 1 1 ′′ ′′ ′′ 4 − 2 2 3 2 k0−k − 2 1 − 2 1 − i τ ( iω + Dq )K (2π) (2π) 2π 0 (ϵ1 k0) + 2 (ϵ2 k ) + 2 ϵ1 k + sinh 2T 4τ 0 4τ 0 2τ 1 1 × F (ω, q, k′′, k′ ) − − i 2 4 − ′′ 2 0 0 ϵ2 k0 2τ D k3 + (k0 k0 )

213 1 1 ≈ δ(k − k′ ) g(0)τ 2 (−iω + Dq2) 0 0 ∫ ∫ 2 ∞ ∞ ′′ − ′′ 2 4e dk0 k0( k0 ) k3 ′′ ′ + dk3 ′′ ′′ F (ω, q, k0 , k0) − 2 2 k0−k 2 4 − 2 π( iω + Dq )K 0 −∞ 2π 0 D k3 + (k0 k0 ) sinh 2T 1 1 ≈ δ(k − k′ ) g(0)τ 2 (−iω + Dq2) 0 0 √ ∫ 2 ′′ − ′′ e 2 dk0 k0( k0 ) √ 1 ′′ ′ + ′′ F (ω, q, k0 , k0). (9.28) − 2 3/2 2 k0−k ′′ ( iω + Dq )D K 2π 0 |(k0 − k )| sinh 2T 0

This gives the matrix equation I/(g(0)τ 2) F = √ , (9.29) (−iω + Dq2)I − e2 2 A D3/2K2 1 where the matrix elements of A1 are given by the integral kernel in the last line of the above equation. As was the case with A0, the largest positive eigenvalue of A1 comes from an eigenvector with constant entries.

We thus obtain the net growth exponent after taking into account the dissipative self-energy:

√ √ e2T 3/2(4 − 2)ζ(3/2) e2T 3/2(5 − 3 2)ζ(3/2) e2T 3/2 λ(1) = √ + ΣR(0, 0) = √ ≈ 1.116 . (9.30) L πD3/2K2 L πD3/2K2 D3/2K2

Hence ∫ ′ 2 dk0 dk0 ′ g(0) 1 f(ω, q) = (g(0)τ) F (ω, q, k0, k0) = . (9.31) 2π 2π (2π)2 − 2 − (1) iω + Dq λL

This returns Eq. (9.2) after a Fourier transform.

9.4 Additional considerations

In the previous section, we computed the squared anticommutator and the leading O(e2) correction to the growth exponent by doing an infinite resummation of the disorder lines. It is natural to ask the following questions: (i) Do ladder diagrams with a different skeleton structure of the disorder lines affect the exponent?

(ii) What is the contribution of the other diagrams at O(e2) that have been ignored in Figure 9.5 above? (iii)

214 How sensitive are the above results to the specific form of the (Coulomb) interaction, V (|r − r′|)?

We address all of these concerns one by one in this section.

9.4.1 Crossed disorder rungs

Instead of using the ‘diffuson’ rung, L(ω, q), considered thus far, we can sum diagrams with ‘maximally- crossed’ disorder rungs (Figure 9.6). As is well known, this gives

1 L (ω, Q) = , (9.32) c g(0)τ 2(−iω + DQ2) where Q is the total momentum of the incoming or outgoing particle-particle pairs. As with L(ω, q), ω is still the net lateral frequency transfer above as the disorder rungs cannot transfer frequency. At the non- interacting level, this gives

fc(ω, 0) ∫ d3k d3k′ dk 1 = 0 GR(k , k)GA(k − ω, k)GR(k , k′)GA(k − ω, k′). (2π)3 (2π)3 2π g(0)τ 2(−iω + D(k + k′)2) 0 0 0 0 0 0 0 0

(9.33)

It is easy to see that this expression does not have a pole at small ω, and hence we do not need to consider contributions with Lc as the base unit (In two spatial dimensions, there is a logarithmic singularity at small

ω that is still weaker than the pole in the contribution with L). We can also insert Lc as an internal rung in

− ′′ − → ′′ − the series with L as the base unit, such as by replacing L(k0 k0 , k2 k3) Lc(k0 k0, k1 + k3) in the integrand of Eq. (9.27). However, in this case, the same small momentum then does not appear in both the interaction and Lc rungs, and the resulting contribution is thus less singular than the one in Eq. (9.27), scaling as subleading powers of T starting at T 2.

Similarly, we can consider insertions such as those in Figure 9.5, but with additional internal L rungs.

215 Figure 9.6: A diagram in the ‘maximally-crossed’ series. The sum of this series gives Lc(ω, Q) as discussed in the main text.

Figure 9.7: Ladder insertions at O(e2), in addition to the ones shown in Figure 9.5, that do not change the growth exponent, λL. The diagrams (a) and (b) have partner diagrams generated by reflection about the horizontal axis. The diagrams (c) and (d) have two partners each, from reflection about the horizontal and vertical axes. Other diagrams (not shown) similar to (c) and (d) with the internal resummed disorder lines terminating on the same time fold instead of opposite time folds vanish due to integrations over Green’s functions with poles on the same side of the real axis.

These are also less singular than the ones shown for the same reason.

9.4.2 Additional diagrams at O(e2)

At O(e2) we have to also consider the diagrams shown in Figure 9.7. In the diagrams given by Figure 9.7(a),

(b), the internal interaction line carries only the external frequency and momentum. We assume that the

Coulomb interaction actually has a long static screening length ξ ≫ l, where l = vF τ is the disorder mean

free path, and that we are probing scrambling at length scales x ≫ ξ.

216 The interaction line in Figure 9.7a is given by

2 2 2 A 4πe (iω + Dq ) 4πe Vξ (ω, q) = lim − = − , (9.34) q→0 q2 + ξ 2 2 q2 2 ξ 2 iω + D K q2+ξ−2 + q

The insertion in Figure 9.7(a) in the limit of small external frequency and momentum (ω, q) is then given by

∫ 2 2 πe dϵ1 dϵ2 1 1 1 1 1 Nig(0) 2 −2 ′′ 2 2 2 2 τ ξ 2π 2π cosh(βk0/2) cosh(βk0 /2) ϵ1 + 1/(4τ ) ϵ1 + i/(2τ) ϵ2 + 1/(4τ ) ϵ2 + i/(2τ) 2 2 2 πe 1 = ig(0) τ −2 ′′ . (9.35) ξ cosh(βk0/2) cosh(βk0 /2)

The factor of i comes from (−i)3 from the three advanced Green’s functions, and the additional factor of N arises because the flavor indices on the left and the right sides of the diagram are decoupled. The partner insertion obtained by reflection about the horizontal axis is the complex conjugate of this, so their sum vanishes. For the insertion in Figure 9.7(b), the internal Wightman line is given by

2 2 2 q 4πe ω DK 2 −2 V W (ω, q) = lim ( ) ( q +ξ ) = 0, (9.36) ξ → 2 −2 2 q 0 q + ξ sinh βω 2 2 2 q2 2 2 ω + D K q2+ξ−2 + q so this diagram is not important. In Figure 9.7 (c), the internal Wightman line carries the external frequency

≲ ≪ 4πe2 ω λL T , so it can be approximated by DK2p2 T , where p is an internal momentum. However, in this case, once again, the same small momentum p does not appear in both the interaction and the internal L or

2 Lc, so this diagram ends up being less singular and scales as subleading powers of T starting at T . For

− 4πe2 Figure 9.7 (d), the internal interaction line is just i K2 and we get for the insertion, after appropriately

217 shifting momenta, for both the internal L and internal Lc cases

∫ Nig(0)πe2 dϵ dϵ d3k dk′′ 1 1 1 2 0 ( ) 4 2 3 ′′ 2 τ K 2π 2π (2π) 2π cosh(βk0/2) cosh(βk /2) 2 1 0 ϵ1 + 4τ 2 × 1 1 1 1 i 2 1 i 2 − − ′′ ϵ1 + ϵ2 + 2 ϵ2 + Dk i(k0 k0 ) 2τ 4τ ∫ 2τ 3Nig(0)πe2τ 2 d3k dk′′ 1 1 = − 0 . (9.37) 2 3 ′′ 2 − − ′′ K (2π) 2π cosh(βk0/2) cosh(βk0 /2) Dk i(k0 k0 )

Reflecting this insertion about the horizontal axis produces its complex conjugate, and reflection about the

′′ vertical axis effectively interchanges k0, k0 . The four contributions then sum to zero.

9.4.3 Short-range interactions

Based on the analysis of Section 9.3, we see that the Lyapunov exponent is simply given by

∫ ∫ d ∞ R (1) − 2 d k dk0 Im[V (k0, k)] λL = 2g(0)τ d Re[L(k0, k)] (2π) −∞ 2π sinh(βk0/2) ∫ ∫ d ∞ R 2 d k dk0 Im[V (k0, k)] + 4g(0)τ d L(k0, k) (2π) −∞ 2π sinh(βk0) ∫ ∫ d ∞ R − 2 d k dk0 Im[V (k0, k)] = 2g(0)τ d L(k0, k) (2π) −∞ 2π sinh(βk0/2) ∫ ∫ d ∞ R 2 d k dk0 Im[V (k0, k)] + 4g(0)τ d L(k0, k), (9.38) (2π) −∞ 2π sinh(βk0)

R as Im[V (k0, k)] and Im[L(k0, k)] are both odd functions of k0 for the interactions we consider. Since

|1/ sinh(βk0/2)| > |2/ sinh(βk0)|, the first term of the above (coming from the ladder sum of Figure 9.5) always dominates the second (coming from the self-energy corrections), and the exponent is thus always

R positive if sgn(Im[V (k0, k)]) = −sgn(k0). For a short-range interaction that does not vanish as q → 0

218 R (we take a contact interaction for which Vbs (q) = V0), screening by the diffuson produces

( ) −iω + Dq2 ω(D − D′)q2 dn V R(ω, q) = V , Im[V R(ω, q)] = V ,D′ = D 1 + V > D. (9.39) s 0 −iω + D′q2 s 0 ω2 + D′2q4 dµ 0

(1) ∼ 2 3/2 Inserting this into Eq. (9.38), we see that all the integrals converge, and that λL +V0 T for d = 3.

Thus, short-range interactions behave qualitatively in the same way as Coulomb interactions from the point of view of scrambling, consistent with previous work on the inelastic scattering rate [317].

9.5 Two spatial dimensions

In two spatial dimensions, the diffuson-screened Coulomb interaction is [317]

2 − 2 R 2πe iω + Dq 2 dn V2 (ω, q) = 2 ,K2 = 2πe . (9.40) q −iω + DK2q + Dq dµ

−1 We probe scrambling at length scales x much larger than the mean free path l and the screening length K2 but smaller than the eventual localization [341] length l ekF l of the electron wavefunctions [317] (The light- cone like growth of f(t, x) will be arrested beyond this localization length, i.e. the operator-radius R(t) is bounded by this length). Then, the same approximations and lines of reasoning we used in three dimensions also work in two dimensions, and the Lyapunov exponent is still given by Eq. (9.38) with d = 2. Inserting this dynamically screened Coulomb interaction, we obtain the leading contribution

∫ ( ) 2 ∞ 2 (1) e 1 − 2 e T T λL2 = dk0 = ln 2 = ln 2 2DK2 0 sinh(βk0/2) sinh(βk0) DK2 2πD(dn/dµ) 2 e R□ k T ≈ B ln 2, (9.41) h ℏ

219 2 where R□ = 1/(e D(dn/dµ)) is the sheet resistivity [247] and we restored factors of kB and ℏ. This cannot

2 saturate the universal bound λL ≤ 2πkBT/ℏ unless the effective coupling e R□/h becomes large, which also determines crossover or transition to an insulating state. According to experimental results reported in

Ref. [342] and theory discussed in Ref. [343], the density-tuned metal-insulator crossover/transition occurs

2 at around R□ ≈ 3h/e , which is smaller than the value required to saturate the bound by about a factor of 3.

This indicates that the metallic state has a Lyapunov exponent numerically, but not parametrically, smaller than the bound.

From Eq. (9.41) above, we see that it contains the difference of two terms. The term being subtracted is the decay rate of electrons in exact eigenstates of the disorder potential [338, 339], whereas the term being added gives the rate at which chaos spreads, i.e. how electrons would be infected within an epidemic picture (See

Figure 9.1) if there were no electron ‘deaths’. Both these terms individually contain a logarithmic infrared divergence, which cancels when their difference is taken. The logarithmic divergence in the exact eigenstate decay rate was removed in a self-consistent computation [340], by using the rate itself as an infrared energy cutoff, but this is not required here. For the exact eigenstate decay rate, the self-consistent computation provides instead a regularized logarithmic factor of ln(πD(dn/dµ)) [247, 317, 340], which doesn’t appear in the Lyapunov exponent.

Let us comment now on why the logarithmic divergence cancels out in the expression for the Lyapunov exponent but appears in the exact eigenstate decay rate. It arises from an infrared divergence in the collision integral in Eq. (9.11) when the energy transfer in a collision approaches zero. At zero energy transfer, the interaction of the electrons with another particle-hole excitation (or equivalently the boson representing the

Coulomb interaction) is like the electrons scattering off a random static potential. Each instance of such a scattering event can be described by a quadratic integrable Hamiltonian, and is hence incapable of producing chaos. However, this process still leads to decoherence of the individual electron wavepackets and hence contributes to the decay rate. A similar cancellation between singular pieces of self-energy and ladder contri-

220 butions coming from zero energy transfer collisions was first pointed out in the computation of the Lyapunov exponent of a Fermi surface coupled to a gapless fluctuating gauge field in Ref. [286]. For the short-range interactions considered in the previous section, the logarithmic factor still cancels in the Lyapunov exponent,

(1) ∼ 2 and we obtain λL2 +V0 T .

9.6 Discussion

We have studied the spread of many-body quantum chaos due to electron-electron interactions in diffusive metals. We find that chaos spreads ballistically, even though quasiparticles are transported diffusively. This is because the spread of chaos is linked only to the propagation of quantum information about inelastic collisions of quasiparticles, which does not require the transport of quasiparticles themselves. In three dimensions, we found that the Lyapunov exponent scales as the inelastic scattering rate of quasiparticles, whereas in two dimensions the inelastic scattering rate is larger than the Lyapunov exponent by a logarithmic factor arising from ‘classical’ collisions that do not involve quantum fluctuations. In d spatial dimensions, we find

d/2 d/4 λL ∼ T , which leads to vB ∼ T . Comparing the form of the butterfly velocity to a scaling form

1−1/z vB ∼ T , where z is the dynamical exponent, we find that our result is qualitatively similar to that of a critical system with z > 1. While our computations in d = 2 and 3 were carried out with the 1/r Coulomb interaction, we expect similar results to hold in d = 2 for the ln r Coulomb interaction.

Remarkably, we find the above ballistic growth of operators even though the Coulomb interaction is long ranged and no microscopic Lieb-Robinson bound exists. This result is a particularly striking example of the idea that the butterfly velocity can function like a low energy Lieb-Robinson velocity [56]. It raises the question of what other long range models might be harboring an emergent ballistic growth of operators at low energy.

We note the recent experimental measurement by Kapitulnik et. al. of local thermal diffusivity using an

221 optical method [18]. It would be interesting to measure the local heat diffusion constant in an interacting diffusive metal using this method. The heat diffusion constant is given by the ratio of thermal conductivity and specific heat; at low enough temperatures, in a regime where both of these quantities are dominated by the electronic contribution, it would be interesting to compare the measured diffusion constant to the known quasiparticle diffusion constant D that appears to be relevant to quantum chaos. While in the non-interacting case one expects the thermal diffusivity to be equal to D, significant deviations may arise due to interactions, especially in two dimensions [344].

In this chapter we only focused on disorder averaged correlation functions in the diffusive, ergodic phase.

However, one could also ask about rare-region effects [345, 346]. For example, can rare ‘localized’ regions in the ergodic phase impede the spread of chaos? How does the different inelastic scattering rate in these regions [347] affect the Lyapunov exponent? Alternatively could there be rare-regions, with very little disorder, that lead to an even faster butterfly velocity? In dimensions greater than one, the effect of such rare-regions are expected to be significantly suppressed, but we leave a detailed study for future work. Finally, it would also be interesting to study the growth of entanglement in an interacting diffusive metal, and compare it to the spread of chaos. We also leave this question for future study.

222 A Appendices to Chapter 1

A.1 Computation of ⟨JyJy⟩

All computations in this appendix are symbolically described for the n = 1 fermions, with the identical contributions for n = 3 accounted for by doubling the overall prefactors. Momentum integrals in dimensional regularization are performed using the standard identity

∫ ka πd/2 Γ((a + d)/2)Γ(b − (a + d)/2) ddk = ∆(a+d)/2−b. (A.1) (k2 + ∆)b Γ(d/2) Γ(b)

223 A.1.1 Free fermion contribution to ⟨JyJy⟩

The free fermion contribution to ⟨JyJy⟩ is given by Fig. 1.2(a) and is straightforwardly computed in dimensional regularization:

⟨JyJy⟩free(ω) = ∫ [ 2 2−ϵ · 2 d k d K Γ K + γd−1ε1(k) − − − − 2(1 + v )NcNf 2 2−ϵ Tr iγd 1( i) 2 2 iγd 1 (2π) (2π) K + ε1(k) ] · × − Γ (K + W) + γd−1ε1(k) ( i) 2 2 (K + W) + ε1(k) √ ∫ ∫ ∫ − 1 dε (k) d2 ϵK −K · (K + W) + ε (k)2 − 2 1 1 = 4NcNf 1 + v dk∥ dx 2 2−ϵ 2 2 2 2 (2π) (2π) [(K + xW) + ε1(k) + x(1 − x)W ] ∫ ∫ 0 ∫ ( ) √ 1 2−ϵ 2 2 NcNf d K −K + W x(1 − x) 1 = − 1 + v2 dk∥ dx + 2−ϵ 2 2 3/2 2 2 1/2 2π 0 (2π) [K + x(1 − x)W ] [K + x(1 − x)W ] ∫ ∫ ( 1−ϵ/2 √ 1 − − NcNf π 1−ϵ Γ(1 ϵ/2)Γ(ϵ/2 1/2) 1/2−ϵ/2 = − 1 + v2 dk∥ ω dx (x(1 − x)) πΓ(1 − ϵ/2)(2π)2−ϵ 2Γ(1/2) 0 ) Γ(1 − ϵ/2)Γ(1/2 + ϵ/2) Γ(2 − ϵ/2)Γ(−1/2 + ϵ/2) + (x(1 − x))1/2−ϵ/2 − (x(1 − x))1/2−ϵ/2 2Γ(3/2) 2Γ(3/2) √ ∫ ( ) 1−ϵ 1 = − 1 + v2 dk∥ N N ω + O(ϵ) , (A.2) c f 16π

where k∥ is the component of k along the Fermi surface.

A.1.2 Fermion self-energy correction to ⟨JyJy⟩

In this subsection, we will freely take the limit of vanishing velocities associated with Eq. (1.26). The extension to the case of finite velocities will be presented in Appendix A.3.

This self-energy correction is given by Fig. 1.2(b) and a partner diagram with the boson on the lower

224 fermion line. The sum of the two gives

⟨JyJy⟩SE(ω) ∫ [ 2 2−ϵ d k d K Γ · K + γd−1ky Γ · K + γd−1ky = 4iN Tr γ − Σ (K, k) f (2π)2 (2π)2−ϵ d 1 K2 + k2 1 K2 + k2 ] y y Γ · (K + W) + γd−1ky × − γd 1 2 2 (A.3) (K + W) + ky

We first compute the fermion self energy, given by Fig. 1.3(a):

Σ1(K, k)

2− ∫ 2 ϵ N∑c 1 2 2−ϵ · − g µ j j d q d Q Γ (K + Q) γd−1(qy + ky) 1 = (τ τ ) iγ − (−i) iγ − N (2π)2 (2π)2−ϵ d 1 (K + Q)2 + (q + k )2 d 1 Q2 + c2q2 f j=1 y y 2− ∫ ∫ 2 ϵ N∑c 1 1 2 2−ϵ g µ d q d Q −Γ · K(1 − x) − γ − (q + k ) = i (τ jτ j) dx d 1 y y N (2π)2 (2π)2−ϵ [Q2 + x(1 − x)K2 + c2q2(1 − x) + x(q + k )2]2 f j=1 0 y y N 2−1 ∫ π1−ϵ/2Γ(1 + ϵ/2) g2µϵ ∑c 1 = i (τ jτ j) dx (2π)2−ϵ N f j=1 0 ∫ 2 − · − − × d q Γ K(1 x) γd−1(qy + ky) 2 2 2 2 2 1+ϵ/2 (2π) [x(1 − x)K + c q (1 − x) + x(qy + ky) ] 2− ∫ ∫ 3/2−ϵ/2 2 ϵ N∑c 1 1 π Γ(1/2 + ϵ/2) g µ dq −Γ · K(1 − x) − γ − (q + k ) = i (τ jτ j) dx √ y d 1 y y , (2π)4−ϵ cN 1 − x [x(1 − x)K2 + x(q + k )2]1/2+ϵ/2 f j=1 0 y y

(A.4)

where in the last step we integrated out qx and then sent c → 0. After shifting qy by ky we integrate it out to get

√ N 2−1 ∫ π2−ϵ/2Γ(ϵ/2) g2µϵ ∑c 1 1 − x Γ · K Σ (K, k) = −i (τ jτ j) dx(x(1 − x))−ϵ/2 . (A.5) 1 (2π)4−ϵ cN x [K2]ϵ/2 f j=1 0

225 Inserting this into the expression for ⟨JyJy⟩SE,

⟨JyJy⟩SE(ω) ∫ π5/2−ϵ/2Γ(ϵ/2)Γ(1/2 − ϵ/2) g2µϵ = 16(1 − N 2) dk c 21−ϵ(2π)8−2ϵΓ(1 − ϵ/2) c x ∫ K4 + W · KK2 − k2(3K2 + K · W) × dk d2−ϵK y y 2 2 2 2 2 2 ϵ/2 (K + ky) ((K + W) + ky)(K ) ∫ ∫ π5/2−ϵ/2Γ(ϵ/2)Γ(1/2 − ϵ/2) g2µϵ 1 = 32(1 − N 2) dk dy(1 − y)× c 21−ϵ(2π)8−2ϵΓ(1 − ϵ/2) c x ∫ 0 d2−ϵK K4 + W · KK2 − k2(3K2 + K · W) × dk y y 2 ϵ/2 2 2 − 2 3 (K ) [(K + yW) + W y(1 y) + ky] ∫ ∫ π7/2−ϵ/2Γ(ϵ/2)Γ(1/2 − ϵ/2) g2µϵ 1 = 4(1 − N 2) dk dy(1 − y)× c 1−ϵ 8−2ϵ − x [ 2 (2π) Γ(1 ϵ/2) c 0 ] d2−ϵK K4 + W · KK2 3K2 + K · W × 3 − (K2)ϵ/2 [(K + yW)2 + W2y(1 − y)]5/2 [(K + yW)2 + W2y(1 − y)]3/2 ∫ ∫ ∫ [ 7/2−ϵ/2 − 2 ϵ 1 1 − 2 π Γ(1/2 ϵ/2) g µ − 2−ϵ = 4(1 Nc ) 1−ϵ 8−2ϵ dkx dz dy(1 y)d K 2 (2π) Γ(1 − ϵ/2) c 0 0 K4 + W · KK2 Γ(5/2 + ϵ/2) 3 z3/2(1 − z)ϵ/2−1− 2 2 − 2 2 − 5/2+ϵ/2 Γ(5/2) [(K + yzW) + y z(1 z)W + W yz(1 y)] ] 3K2 + K · W Γ(3/2 + ϵ/2) − z1/2(1 − z)ϵ/2−1 . (A.6) [(K + yzW)2 + y2z(1 − z)W2 + W2yz(1 − y)]3/2+ϵ/2 Γ(3/2)

Now we shift K → K − yzW. This leads to the replacements ((K · W)2 ≡ K2W2/(2 − ϵ) as far as integration over K is concerned)

4K2W2y2z2 2K2W2yz K4 + W · KK2 → K4 + + 2K2W2y2z2 − − 2 − ϵ 2 − ϵ

− K2W2yz + W4y4z4 − W4y3z3,

K2 → K2 + W2y2z2,

K · W → −yzW2. (A.7)

226 Then, integrating out K,

∫ ∫ ∫ π9/2−ϵΓ(1/2 − ϵ/2) g2µϵ 1 1 ⟨J J ⟩ (ω) = 4(1 − N 2) ω1−2ϵ dk dz dy y y SE c −ϵ 8−2ϵ − 2 x [ 2 (2π) Γ(1 ϵ/2) c 0 0 × (1 − y)(1 − z)ϵ/2−1× ( ( )( ( ) ( )) ( )− − 3y2z2 − yz Γ ϵ + 1 Γ 1 − ϵ y2(1 − z)z + (1 − y)yz ϵ 1/2 z1/2 × − 2 2 ( ) 3 + 2Γ 2 ( ( ) ( )) ( ( ) ( )) ( ) 1 ϵ 4 2 2 −ϵ−1/2 3/2 3yz Γ ϵ + Γ 2 − yz − + 2 − − + 1 y (1 − z)z + (1 − y)yz z 2 2 2 ϵ 2( ϵ) + 5 2Γ 2 ( ( ) ( )) ( ) − 3 Γ ϵ − 1 Γ 2 − ϵ y2(1 − z)z + (1 − y)yz 1/2 ϵ z1/2 − 2 2 ( ) 3 + 2Γ 2 ( ( ) ( )) ( ) − 3 Γ ϵ − 1 Γ 3 − ϵ y2(1 − z)z + (1 − y)yz 1/2 ϵ z3/2 + 2 2 ( ) + 2Γ 5 2 )] ( ( ) ( )) ( )− − 3y3z3(yz − 1) Γ ϵ + 3 Γ 1 − ϵ y2(1 − z)z + (1 − y)yz ϵ 3/2 z3/2 2 2 ( ) + 5 . (A.8) 2Γ 2

To leading order in ϵ, we can take only the (1 − z)ϵ/2−1 term in the above integrand for its z dependence and set z = 1 elsewhere (which produces 2/ϵ for the integral over z). This agrees with numerically evaluating the y and z integrals. We thus get,

∫ ∫ √ (1 − N 2) g2µϵ 1 1 − y ⟨J J ⟩ (ω) = dk c ω1−2ϵ dy y(6y − 5) y y SE x 32π3ϵ c y ∫ ( 0 ) g2µϵ 1 = dk (N 2 − 1) ω1−2ϵ + O(1) . (A.9) x c c 128π2ϵ

A.1.3 Vertex correction to ⟨JyJy⟩

As in Appendix A.1.2, here too we will freely take the limit of vanishing velocities associated with Eq. (1.26).

The case of finite velocities will be presented in Appendix A.3.

227 This correction is then given by Fig. 1.2(c):

∫ ∫ [ 2 2−ϵ d k d K Γ · K + γd−1ky ⟨J J ⟩ (ω) = 2iN Tr γ − Ξ (K, k, W) y y vert f (2π)2 (2π)2−ϵ d 1 K2 + k2 3 ] y · × Γ (K + W) + γd−1ky 2 2 . (A.10) (K + W) + ky

We again first compute the current (Jy) vertex, given by Fig. 1.3(b):

Ξ3(K, k, W)

2− ∫ 2 ϵ N∑c 1 2 2−ϵ · − g µ j j d q d Q Γ (K + Q) γd−1(qy + ky) = (τ τ ) iγ − × (−i) (−iγ − ) N (2π)2 (2π)2−ϵ d 1 (K + Q)2 + (q + k )2 d 1 f j=1 y y

Γ · (K + Q + W) − γd−1(qy + ky) 1 × − − ( i) 2 2 iγd 1 2 2 2 (K + Q + W) + (qy + ky) Q + c q N 2−1 ∫ ∫ ∫ [ g2µϵ ∑c 1 1−x d2q d2−ϵQ = −2i (τ jτ j) dx dy − (K + Q) · (K + Q + W) N (2π)2 (2π)2−ϵ f j=1 0 0

] 2 ¯ ¯ + (qy + ky) − 2Γ · (K + Q)γd−1(qy + ky) + Γ · Wγd−1(qy + ky) − Γ · (K + Q)Γ · W × [ × (Q + (x + y)K + yW)2 + W2y(1 − y) + (1 − (x + y))(K2(x + y) + 2K · Wy)+ ] −3 2 2 2 + (x + y)(qy + ky) + (1 − (x + y))q c γd−1 ( N 2−1 ∫ ∫ ∫ g2µϵπ3/2−ϵ/2 ∑c 1 1−x 1 = −i (τ jτ j) dx dy √ dq cN (2π)4−ϵΓ(1 − ϵ/2) − y f j=1 0 0 (1 (x + y)) [ 2 (ky + qy) − (K(x + y − 1) + W(y − 1)) · (K(x + y − 1) + Wy)+ ] ¯ ¯ 2Γ · (K(1 − (x + y)) − Wy)γd−1(ky + qy) − Γ · K(1 − (x + y))Γ · W + Γ · Wγd−1(ky + qy) ) × − −(3/2+ϵ/2) − − −(1/2+ϵ/2) Γ(3/2 + ϵ/2)Γ(1 ϵ/2)∆1 Γ(1/2 + ϵ/2)Γ(2 ϵ/2)∆1 γd−1, (A.11)

228 where we again integrated out qx and then sent c → 0 in the last step of the above, and

2 2 2 ∆1 = W y(1 − y) + (1 − (x + y))(K (x + y) + 2W · Ky) + (x + y)(ky + qy)

Proceeding,

( 2− ∫ ∫ 2 ϵ 2−ϵ/2 N∑c 1 1 1−x − g µ π j j √ 1 Ξ3(K, k, W) = i 4−ϵ (τ τ ) dx dy cNf (2π) Γ(1 − ϵ/2) 0 0 (1 − (x + y))(x + y) [ j=1 ] − (K(x + y − 1) + W(y − 1)) · (K(x + y − 1) + Wy) − Γ¯ · K¯ (1 − (x + y))Γ · W × ( ) Γ(1 − ϵ/2)Γ(1 + ϵ/2) Γ(ϵ/2) Γ(1 − ϵ/2) × − Γ(2 − ϵ/2) − γ − , (A.12) (1+ϵ/2) ϵ/2 2(x + y) d 1 ∆2 ∆2 where

2 2 ∆2 = W y(1 − y) + (1 − (x + y))(K (x + y) + 2W · Ky). (A.13)

An important feature of the above computation is that because

∫ ∫ 1 1−x 2(x + y) − 1 dx dy √ = 0, (A.14) 3/2 0 0 (x + y) 1 − (x + y) the coefficient of the 1/ϵ pole (i.e. the coefficient of Γ(ϵ/2)) in Ξ vanishes when ϵ → 0. This eventually leads to the lack of a 1/ϵ pole in ⟨JyJy⟩vert, and hence the correction to scaling of ⟨JyJy⟩ arises solely from the self-energy graphs.

229 Taking the expression for the current vertex and inserting it into the one for ⟨JyJy⟩vert, we get

⟨JyJy⟩vert(ω) ∫ ∫ ∫ ∫ ∫ ( 2 ϵ 3−ϵ/2 2 − 1 1 1−x 4g µ π (Nc 1) 2−ϵ dy = dkx d K dz dx √ c(2π)8−2ϵΓ(1 − ϵ/2) (1 − (x + y))((x + y) [ 0 0 (0 ) 1 K · (K + W) − (K(x + y − 1) + W(y − 1)) · (K(x + y − 1) + Wy) − + ∆1/2 ∆3/2 ] ( 3 3 K¯ 2W2(1 − (x + y)) Γ(1 − ϵ/2)Γ(1 + ϵ/2) Γ(ϵ/2) + − Γ(2 − ϵ/2)− ∆3/2 ∆(1+ϵ/2) ∆ϵ/2 3 )( 2 )) 2 Γ(1 − ϵ/2) 1 K · (K + W) − − , (A.15) 2(x + y) 1/2 3/2 ∆3 ∆3 where now

2 2 ∆2 = W y(1 − y) + (1 − (x + y))(K (x + y) + 2W · Ky),

2 2 ∆3 = (K + zW) + z(1 − z)W . (A.16)

We combine denominators using

∫ 1 Γ(s + b) 1 as−1(1 − a)b−1 = da , (A.17) s b − s+b ∆2∆3 Γ(s)Γ(b) 0 [a∆2 + (1 a)∆3] and the denominator square completion is

( ) W(ay(1 − (x + y)) + (1 − a)z) 2 a∆ + (1 − a)∆ = (a(x + y)(1 − (x + y)) + (1 − a)) + K 2 3 a(x + y)(1 − (x + y)) + (1 − a) ( ) (ay(1 − (x + y)) + (1 − a)z)2 + W2 − + a(1 − y)y + (1 − a)z . (A.18) a(x + y)(1 − (x + y)) + (1 − a)

230 Defining

f1 = a(x + y)(1 − (x + y)) + (1 − a), 1 f = (ay(1 − (x + y)) + (1 − a)z), f1

2 f2 = a(1 − y)y + (1 − a)z − f1f , (A.19)

we can process the numerators and write down the final expression

∫ ∫ ∫ ∫ ∫ ( ) 2 ϵ 4−ϵ 2 − 1 1 1 1−x ⟨ ⟩ 8g µ π (Nc 1) 1−2ϵ JyJy vert(ω) = 8−2ϵ 2 ω dkx da dz dx dy T1 + T2 + T3 + T4 , c(2π) Γ(1 − ϵ/2) 0 0 0 0 (A.20)

231 where

( ( √ ϵ −3 −ϵ− 3 ( ) ( ) ( ) − ϵ/2 2 2 1 aa f1 f2 ϵ 1 ϵ 1 2 T1 = √ Γ 1 − f1f2Γ 2 − Γ ϵ + 2f (ϵ − 4) −(x + y − 1)(x + y)π 2 ϵ − 2 2 2

× (x + y − 1)2 − f(ϵ − 4)(x + 3y − 2)(x + y − 1) − 2xy ) + x + (y − 1)(y(ϵ − 4) + 1) − (1 − (x + y))(ϵ − 1)

( ) ( ) − 2 − ϵ 3 − − − − + (f 1)ff1 Γ 1 Γ ϵ + (f(x + y 1) y)(f(x + y 1) y + 1)+ 2 2 ) ( ) ( ) ϵ 1 + f 2(x + y − 1)2Γ 3 − Γ ϵ − , 2 2 2 ( ϵ − − − 1 ( ) ϵ/2 2 2 ϵ 2 ( ) ( ) a f1 f2 ϵ ϵ 1 T2 = √ √ Γ 1 − − f1Γ 1 − Γ ϵ + 1 − a −(x + y − 1))(x + y)π 2 2 2 ) ( ) ( ) ϵ 1 × (f(x + y − 1) − y)(f(x + y − 1) − y + 1) − f (x + y − 1)2Γ 2 − Γ ϵ − , 2 2 2 ( √ ϵ − − − 1 ( ) ϵ −1 2 ϵ ( ) 1 − aa 2 f 2 f 2 ϵ 2 1 T = √ √ 1 2 Γ 1 − Γ ϵ − (x(ϵ − 2) + y(ϵ − 2) + 1) f 2× 3 4 π −x − y + 1(x + y)3/2 2 2 )

× (f1 − 2f1ϵ) + ff1(2ϵ − 1) + f2(ϵ − 2) ,

ϵ ϵ −1 1 −ϵ ( ) ( ) −1 2 2 2 a 2 f1 f2 ϵ 1 T4 = √ √ Γ 1 − Γ ϵ − (x(ϵ − 2) + y(ϵ − 2) + 1). (A.21) 4 1 − a π(−x − y + 1)(x + y)3/2 2 2

This multidimensional integral over four parameters is finite in the limit of ϵ → 0 and can be done numerically. We first integrate over x and y: The resulting function of a and z has integrable singularities in the limits of a → 1 and a → 0 which can be handled by numerical integration using an adaptive grid. The final result is ∫ ( ) g2µϵ(N 2 − 1) ⟨J J ⟩ (ω) = dk c ω1−2ϵ α + O(ϵ) , (A.22) y y vert x 32π4c 0

where α0 ≈ 1.1 is a finite numerical constant.

232 A.2 Free Energy Computations

As in the previous appendix, we will freely take the limit of vanishing velocities associated with Eq. (1.26) here as well to compute the correction to the fermion free energy.

  N 2−1 ∫ 1 ∑c d2qd1−ϵQ¯ ∑ 1 F = Tr  τ jτ j T [(Π(q, T ) − Π(q, 0)) + Π(q, 0)] . fb 2 (2π)3−ϵ Q¯2 + c2|q|2 + ω2 j=1 ωq q (A.23)

Where Π(q, T ) is the fermion RPA bubble at external momentum and frequency given by q evaluated at temperature T . As described in the main text, we evaluate the finite temperature part of the bubble at v = 0 to renormalize the fermion free energy and the zero temperature part at v ≠ 0 to renormalize the boson free energy. To evaluate the frequency summations, we use the following zeta-function regularization identities:

∑ 1 T 1−s T s = 2 s ζ(s), |ωq| (2π) ωq ( ) ∑ 1 T 1−s 1 T s = 2 s ζ s, . (A.24) |ωk| (2π) 2 ωk

Where ωq is a bosonic Matsubara frequency and ωk is a fermionic Matsubara frequency. Note that the first line of Eq. (A.24) ignores a formally infinite term coming from s = 0, however, as will be evident in the below computations, this infinite term never multiplies quantities that are ∝ 1/ϵ, and hence does not

233 renormalize the scaling of the free energy. We then have

∫ ∫ ( ∫ ) 1−ϵ ¯ ∑ 2 dkx dkyd K dωk Π(q, T ) − Π(q, 0) = −4g T − Tr [iγ − G(k)iγ − G(k + q)] 2π (2π)2−ϵ 2π d 1 d 1 ωk ∫ ∫ ( ∫ ) dk d1−ϵK¯ ∑ dω = −4g2 dk y T − k × x (2π)3−ϵ 2π [ ωk Q¯2 + q2 + ω2 y q − ¯ 2 2 2 ¯ ¯ 2 2 2 (K + k + ω )((K + Q) + (ky + qy) + (ωk + ωq) ) y k ] 1 1 − − . (A.25) ¯ 2 2 2 ¯ ¯ 2 2 2 K + ky + ωk (K + Q) + (ky + qy) + (ωk + ωq)

The last two terms in the square brackets yield identical contributions, because the q in the last term can be shifted out. Thus,

Π(q, T ) − Π(q, 0) ∫ ∫ ( ∫ ) dk d1−ϵK¯ ∑ dω = −4g2 dk y T − k × x (2π)3−ϵ 2π [ ωk ] Q¯2 + q2 + ω2 y q + ¯ 2 2 2 ¯ ¯ 2 2 2 (K + ky + ωk)((K + Q) + (ky + qy) + (ωk + ωq) ) ∫ ∫ ( ∫ ) dk d1−ϵK¯ ∑ dω 1 + 8g2 dk y T − k . x 3−ϵ ¯ 2 2 2 (2π) 2π K + ky + ω ωk k

(A.26)

234 We evaluate the second term to leading order in ϵ in the above using dimensional regularization for the momentum integral and zeta function regularization for the frequency sum:

∫ ∫ ( ∫ ) dk d1−ϵK¯ ∑ dω 1 8g2 dk y T − k x 3−ϵ ¯ 2 2 2 (2π) 2π K + ky + ω ωk k ∫ ∫ dk d1−ϵK¯ ∑ 1 = 8g2 dk y T x 3−ϵ ¯ 2 2 2 (2π) K + ky + ω ωk k ∫ ∫ 2 ∑ 2 1−ϵ 8πg 1 − 8g T ln 2 = 2 dkxT ϵ = dkx 2 , (A.27) (2π) ϵ |ωk| (2π) ωk where we have used the fact that scaleless integrals vanish in dimensional regularization. Thus,

Π(q, T ) − Π(q, 0) ∫ ∫ (∫ ) dk d1−ϵK¯ dω ∑ = 4g2 dk y k − T × x (2π)3−ϵ 2π [ ωk ] Q¯2 + q2 + ω2 y q − ¯ 2 2 2 ¯ ¯ 2 2 2 (K + ky + ωk)((K + Q) + (ky + qy) + (ωk + ωq) ) ∫ 8g2T 1−ϵ ln 2 − dk , (A.28) x (2π)2

To evaluate the first term, we introduce a Feynman parameter y to combine the denominators. Doing the k momentum integral and ωk frequency summation (integral for the T = 0 part), we have, to leading order in

∫ ∫ g2 1 [ ] Π(q, T ) − Π(q, 0) = − dk dy t(y, q˜2, ω , ϵ) − 2π (˜q2 + ω2)1/2−ϵ/2 8π2 x q q ∫ 0 8g2T 1−ϵ ln 2 − dk (A.29) x (2π)2

235 ¯2 2 1/2 Where q˜ = (Q + qy) . We determine the following asymptotic expansion numerically

∫ ( 1 16 ln 2 2ω2 − q˜2 dy t(y, q˜2, ω , ϵ) = 2π − T 1−ϵ + 48ζ(3)T 3−ϵ q q 2 2 1/2−ϵ/2 2 2 5/2−ϵ/2 0 (˜q + ωq ) (˜q + ωq ) ( ) ) T 5−ϵ + O + ... . (A.30) 5−ϵ 5−ϵ q˜ , ωq

Simple power counting dictates that the higher terms in the above asymptotic expansion can’t produce any log UV divergences in the final two loop graph because they fall off too fast in q. Thus, retaining only terms that will survive and contribute to the pole in the final two-loop integral,

∫ ( ) 2 2ω2 − q˜2 − − g 3−ϵ q Π(q, T ) Π(q, 0) = 2 dkx 48ζ(3)T 2 2 2 . (A.31) 8π (˜q + ωq )

We evaluate Π(q, 0) using dimensional regularization at finite v to get, to leading order in ϵ:

∫ 2 2−ϵ 2 d k d K Π(q, 0) = −4g Tr [iγ − G (K, k)iγ − G (K + Q, k + q)] (2π)2 (2π)2−ϵ d 1 n d 1 n¯ ∫ 2 2−ϵ − · −4g dεn(k)dεn¯(k) d K K (K + Q) + εn(k)εn¯(k + q) = 2 2−ϵ 2 2 2 2 v (2π) (2π) [K + εn(k) ][(K + Q) + εn¯(k + q) ] ∫ g2 d2−ϵK −K · (K + Q) = − v (2π)2−ϵ [K2]1/2[(K + Q)2]1/2 2 − g2 Q2−ϵ g2 (Q + ω2)1 ϵ/2 = − = − q , (A.32) v 8πϵ v 8πϵ

236 where the last integral was performed using Feynman parameterization. Inserting the expressions for Π(q, T )−

Π(q, 0) and Π(q, 0) into Eq. (A.23), we get, using dimensional regularization for the q momentum integrals,

(1) (2) Ffb = Ffb + Ffb , ∫ d2qd1−ϵQ¯ ∑ Π(q, T ) − Π(q, 0) F (1) = (N 2 − 1) T fb c (2π)3−ϵ Q¯2 + c2|q|2 + ω2 ωq q ∫ ∫ 2 1−ϵ ¯ ∑ 2ω2 − q˜2 − 2 3−ϵ 6ζ(3)g dqyd Q q √ = (1 Nc )T dkx 3−ϵ T (A.33) πc (2π) 2 2 2 ¯2 2 ωq (˜q + ωq ) Q + ωq

Where we integrated out qx and then sent c → 0 in the last step of the above. Doing the remaining integrals first over qy and then Q¯, we get, to leading order in ϵ

∫ ∫ 2 2 − 3−ϵ ∑ 2 2 − 3−2ϵ (1) 3ζ(3)g (Nc 1)T 1 3ζ(3)g (Nc 1)T Ffb = 2 dkxT 1+ϵ = dkx 3 . (A.34) 16cπ |ωq| 16cπ ϵ ωq

The other part gives

∫ d2qd1−ϵQ¯ ∑ Π(q, 0) F (2) = (N 2 − 1) T fb c (2π)3−ϵ Q¯2 + c2|q|2 + ω2 ωq q ∫ − g2 d2qd1−ϵQ¯ ∑ (Q¯2 + ω2)1 ϵ/2 = (1 − N 2) T q . (A.35) c 8πvϵ (2π)3−ϵ Q¯2 + c2|q|2 + ω2 ωq q

Integrating first over Q¯ and then over q using the dimensional regularization, we get, to leading order in ϵ

g2π ∑ 1 g2π F (2) = (N 2 − 1) T = (N 2 − 1) T 4−2ϵ. fb c 2 −(3−2ϵ) c 2 (A.36) 6vc ϵ |ωq| 360vc ϵ ωq

237 A.3 Finite v and c

In this appendix, we describe the breakdown of the results derived in the previous appendices when we do not have v, c → 0. We illustrate this by first computing the self energy correction to ⟨JyJy⟩ for finite v and c; similar problems occur in the computations of the fermion free energy. The fermion self energy for v, c ≠ 0 is given by [89]

∫ 2− c2ε (k) − N∑c 1 · − 3 2 ϵ/2 2 ϵ 1 Γ K γd−1 2 2 2 − π Γ(ϵ/2) g µ j j c +x(1+v −c ) Σ1(K, k) = i 4−ϵ dx (τ τ ) [ ] (2π) cN c2ε2(k) ϵ/2 f 0 j=1 2 3 K + c2+x(1+v2−c2) x−ϵ/2(1 − x)1/2−ϵ/2 × (A.37) (c2 + x(1 + v2 − c2))1/2

We can ignore the term with the prefactor of c2 in the numerator of the integrand in Eq. (A.37); since v ≠ 0

ε1(k) and ε3(k) can be taken to be independent variables of integration over k space via the coordinate

2 transformation d k → dε1dε3/(2v). This term then only produces contributions to ⟨JyJy⟩SE that are odd in

ε3 and hence vanish under integration over ε3. Thus dropping this term, we have

∫ 16(1 − N 2)π2−ϵ/2Γ(ϵ/2)g2µϵ 1 x−ϵ/2(1 − x)1/2−ϵ/2(1 + v2) ⟨J J ⟩ (ω) = c dx × y y SE 8−2ϵ 2 2 2 1/2 (2π) c 0 (c + x(1 + v − c )) ∫ 4 · 2 − 2 2 · dε1dε3 2−ϵ K + W KK ε1(3K + K W) × d K [ ] ϵ (A.38) 2 2 2v 2 2 2 2 2 2 c ε3 2 (K + ε1) ((K + W) + ε1) K + c2+x(1+v2−c2) evaluating this as in Appendix A.1.2 gives the singular contribution

⟨JyJy⟩SE(ω) ≈ ∫ ∫ ( ) dε g2µϵ 1 c2ε2 3 (N 2 − 1) ω1−2ϵ dxκ 3 , ϵ 2v c c 1 ω2(c2 + x(1 + v2 − c2)) (0 ) (1 − x)1/2(1 + v2) 1 × (A.39) (c2 + x(1 + v2 − c2))1/2 64π3ϵ

238 −ϵ/2 where the crossover function κ1(x, ϵ) ≈ (1 + x) for x ≪ 1.

The singular part of the 1-loop current vertex at finite v and c is most easily derived from the Ward identity;

dΣ1(K, k) Ξ3(K, k, 0) = − = dky pole pole 2− ∫ [ ] 2−ϵ/2 2 ϵ N∑c 1 1 2 2 −ϵ/2 π Γ(ϵ/2) g cµ j j 2 c ε3(k) iγ − (τ τ ) dx K + d 1 (2π)4−ϵ N c2 + x(1 + v2 − c2) f j=1 0 x−ϵ/2(1 − x)1/2−ϵ/2 × (A.40) (c2 + x(1 + v2 − c2))3/2

Inserting this into Eq. (1.29) gives, for the singular part of the two-loop vertex correction to ⟨JyJy⟩, via a computation very similar to that for the self energy correction (an additional prefactor of 2 has to be inserted to account for both the poles associated with vertex corrections to each of the two current vertices in the graph),

∫ ∫ g2cµϵ(N 2 − 1) dε dε 1 (1 − x)1/2(1 − v2) ⟨J J ⟩ (ω) ≈ − c 1 3 d2−ϵK dx × y y vert 8π6ϵ 2v 2 2 − 2 3/2 [ 0 ] (c + x(1 + v c )) −K · (K + W) + ε2 [ 1 ] 2 2 ϵ/2 2 2 2 2 2 c ε3 (K + ε1)((K + W) + ε1) K + c2+x(1+v2−c2) ∫ ∫ ( ) dε 1 c2ε2 ≈ − 3 (N 2 − 1)g2cµϵω1−2ϵ dxκ 3 , ϵ 2v c 2 ω2(c2 + x(1 + v2 − c2)) ( 0 ) (1 − x)1/2(1 − v2) 1 × , (A.41) (c2 + x(1 + v2 − c2))3/2 32π3ϵ

−ϵ/2 where again the crossover function κ2(x, ϵ) ≈ (1 + x) for x ≪ 1.

239 A.4 Boltzmann Equation Computations

A.4.1 Collisionless conductivity in d = 3 − ϵ

We can diagonalize the Hamiltonian corresponding to the free fermion part of Eq. (1.8) as

4 N Nf ∫ ∑ ∑c ∑ d2kd1−ϵK¯ [ ] H = Ψ¯ (k, K¯ ) iΓ¯ · K¯ + iγ − ε (k) Ψ (k, K¯ ) f (2π)3−ϵ n,σ,j d 1 n n,σ,j n=1 σ=1 j=1 N ∫ ∑4 ∑Nc ∑f ∑ 2 1−ϵ d kd K¯ † = λ (k, K¯ )ξ (k, K¯ )λ (k, K¯ ) (A.42) (2π)3−ϵ n,σ,j,m n,m n,σ,j,m n=1 σ=1 j=1 m=

( ) ¯ ¯ 2 2 1/2 with the particle-hole symmetric dispersions ξn,m(k, K) = m K + εn(k) . The physical current density becomes

( ) N ∫ ∑4 ∑Nc ∑f ∑ 2 1−ϵ d kd K¯ mεn(k) † J = vn √ λ (k, K¯ )λn,σ,j,m(k, K¯ ) + J2, (A.43) (2π)3−ϵ ¯ 2 2 n,σ,j,m n=1 σ=1 j=1 m= K + εn(k)

· † † where εn(k) = vn k and J2 contains particle-hole terms λ+λ−, λ−λ+ that are unimportant for transport in the low frequency regime of ω ≪ T [107, 108]. Defining the distribution functions

¯ ⟨ † ¯ ¯ ⟩ fn,m(k, K, t) = λn,σ,j,m(k, K, t)λn,σ,j,m(k, K, t) , (A.44)

we have the collisionless kinetic equation in the presence of an applied electric field

( ) ∂ ∂ + mE · f (k, K¯ , t) = 0, (A.45) ∂t ∂k n,m

240 with the frequency-domain solution to linear order in E

¯ ¯ mεn(k) 1/T fn,m(k, K, ω) = 2πδ(ω)nf (ξn,m(k, K)) + vn · E(ω)√ 2 2 − + K¯ + εn(k) iω + 0

¯ ¯ × nf (ξn,m(k, K))(1 − nf (ξn,m(k, K))). (A.46)

Inserting this into the expression for J, we obtain the collisionless conductivity

∫ δJ (ω) (1 + v2)/T d2kd1−ϵK¯ ε2 (k) σ (ω) = x = 4N N n xx c f − + 3−ϵ ¯ 2 2 δEx(ω) iω + 0 (2π) K + εn(k) ¯ ¯ × nf (ξn,+(k, K))(1 − nf (ξn,+(k, K))), ∫ ∫ √ 1−ϵ ¯ 2 δ(ω) dεnd K εn Re[σ (ω)] = 2N N dk∥ 1 + v2 xx c f 2−ϵ ¯ 2 2 T (2π) K + εn (√ )( (√ )) × ¯ 2 2 − ¯ 2 2 nf K + εn 1 nf K + εn ∫ √ 1−ϵ/2 − ϵ − − 1−ϵ π (1 2 )Γ(2 ϵ)ζ(1 ϵ) = 2N N 1 + v2 dk∥δ(ω)T = Re[σ (ω)]. (A.47) c f (2π)2−ϵΓ(2 − ϵ/2) yy

A.4.2 Derviation of the fermion collision integral

We derive the following expressions for the different components of the fermion self energies in the Keldysh

R formalism. For a fermion at hot spot given by (ℓ, +), we get for the first diagram for Σf in Fig. 1.5

Rℓ+(1) ′ ′ − ′ ′ − ′ 2 a a K Rℓ ′ 2 K Rℓ Σf,σσ′ (x, x ) = ig τσρτρσ′ D0 (x, x )G0 (x, x ) = 3iδσσ g D0 (x, x )G0 (x, x ). (A.48)

We use that for products of Wigner transforms

∑ K ′ Rℓ− ′ → K − Rℓ− D0 (x, x )G0 (x, x ) D0 (x, p q)G0 (x, q), (A.49) q

241 and plug in the representation of the Keldysh propagator in terms of the distribution function to get

∑ [ ] Rℓ+(1) − ′ 2 − R − − A − Rℓ Σf,σσ′ (x, p) = 3δσσ g Fb(x, p q) i D0 (x, p q) D0 (x, p q) G0 (x, q). (A.50) q

For the collision integral on the right hand side of (1.59) we need twice the imaginary part of this expression.

Using

[ ] − 1 − − 2Im[GRℓ (x, q)] = GRℓ (x, q) − GAℓ (x, q) , (A.51) 0 i 0 0 we get,

[ ] Rℓ+(1) 2Im Σf,σσ′ (x, p) ∑ [ ] [ ] 2 R A 1 Rℓ− Aℓ− = 3δ ′ g F (x, p − q) i D (x, p − q) − D (x, p − q) G (x, q) − G (x, q) σσ b 0 0 i 0 0 q ∫ ∫ dω 1 ( ) − ′ 2 2 + − − − + − × = 3δσσ g d q δ(eℓ (p) ω ωp−q) δ(eℓ (p) ω + ωp−q) 2π 4ωp−q

× − − − + − δ(ω eℓ (q))Fb(t, p q, eℓ (p) ω) ∫ 2 ( d q 1 − − ′ 2 + − − − − = 3δσσ g δ(eℓ (p) eℓ (q) ωp−q)Fb(t, p q, ωp−q) 2π 4ωp−q ) − + − − − − δ(eℓ (p) eℓ (q) + ωp−q)Fb(t, p q, ωp−q) [ ] Rℓ+(1) ′ + = 2δσσ Im Σf (t, p, eℓ (p)) , (A.52)

242 where we have used spatial translational invariance and also have kept the external fermion on shell. Like-

R Rℓ+ wise, for the second diagram for Σf in Fig. 1.5 contributing to Σf,σσ′ we have

Rℓ+(2) 2Im[Σf,σσ′ (x, p)] ∑ − − ′ 2 Rℓ − Aℓ R − − A − = 3δσσ g Ff (x, q)[G0 (x, q) G0 (x, q)][D0 (x, p q) D0 (x, p q)] q ∫ 2 ( ) d q 1 − − − − ′ 2 + − − − + − ℓ = 3δσσ g δ(eℓ (p) eℓ (q) ωp−q) δ(eℓ (p) eℓ (q) + ωp−q) Ff (t, q) 2π 4ωp−q [ ] Rℓ+(2) ′ + = 2δσσ Im Σf (t, p, eℓ (p)) . (A.53)

Kℓ+ For the diagrams in Fig. 1.5 contributing to Σf,σσ′ , the first gives

Kℓ+(1) iΣf,σσ′ (x, p) ∑ − − − ′ 2 − Rℓ − Aℓ R − − A − = 3δσσ g Ff (x, q)Fb(x, p q)[G0 (x, q) G0 (x, q)][D0 (x, p q) D0 (x, p q)] q ∫ ( 2 d q 1 − − ′ 2 + − − ℓ − − = 3δσσ g δ(eℓ (p) eℓ (q) ωp−q)Ff (t, q)Fb(t, p q, ωp−q) 2π 4ωp−q ) − + − − ℓ− − − δ(eℓ (p) eℓ (q) + ωp−q)Ff (t, q)Fb(t, p q, ωp−q)

Kℓ+(1) ′ + = 2iδσσ Σf (t, p, eℓ (p)). (A.54)

243 The second and third combined yield

Kℓ+(2+3) iΣf,σσ′ (x, p) ∑ − − − ′ 2 Rℓ R − Aℓ A − = 3δσσ g [G0 (x, q)D0 (x, p q) + G0 D0 (x, p q)] q ∫ ∫ [ ( d2q dω g2 1 1 = 3δ ′ σσ 2 − + + + 4π 2π 4ωp−q ω − e (q) + i0 e (p) − ω + ω − + i0 ) ℓ] ℓ p q 1 − + c.c. + − − + eℓ (p) ω ωp−q + i0 ∫ 2 ( ) d q 1 − − ′ 2 + − − − + − = 3δσσ g δ(eℓ (p) eℓ (q) ωp−q) δ(eℓ (p) eℓ (q) + ωp−q) 2π 4ωp−q

Kℓ+(2+3) ′ + = 2iδσσ Σf (t, p, eℓ (p)). (A.55)

Combining the above expressions gives the collision integral for fermions of any spin at the hot spot given by (ℓ, +)

coll Ifℓ+[Ff ,Fb](t, p) = ( ∫ [ 2 2 1 + − − − ℓ− − = 3g d q δ(eℓ (p) eℓ (q) ωp−q) 1 + Ff (t, q)Fb(t, p q, ωp−q) 4ωp−q

− ℓ− ℓ+ Ff (t, q)Ff (t, p) ] [ − ℓ+ − − + − − ℓ− − − Ff (t, p)Fb(t, p q, ωp−q) δ(eℓ (p) eℓ (q) + ωp−q) 1 + Ff (t, q)Fb(t, p q, ωp−q) ) ] − ℓ− ℓ+ − ℓ+ − − Ff (t, q)Ff (t, p) Ff (t, p)Fb(t, p q, ωp−q) . (A.56)

The collision integral for the hot spot given by (ℓ, -) is given by simply interchanging + ↔ − in the above.

With the alternate parameterization (1.58) of the distribution functions, and relabeling of momenta q ↔ p−q

244 we may rewrite this for any hot spot as

coll Ifℓ[ff , fb](t, p) = ∫ ( 2 [ 2 d q 1 − ∓ − − ℓ − ℓ∓ − = 3g δ(eℓ (p) eℓ (p q) ωq) ff (t, p)(1 ff (t, p q)) 2π ωq

− ℓ∓ − ff (t, p q)fb(t, q, ωq) ] [ ℓ − − ∓ − ℓ − ℓ− − + ff (t, p)fb(t, q, ωq) δ(eℓ (p) eℓ (p q) + ωq) ff (t, p)(1 ff (t, p q)) ) ] − ℓ∓ − − ℓ − ff (t, p q)fb(t, q, ωq) + ff (t, p)fb(t, q, ωq) . (A.57)

A.4.3 Derivation of the boson collision integral

We begin with the retarded component of the boson self energy in the Keldysh formalism. The sum of the two diagrams for this component in Fig. 1.5 gives

∑ [ ] R ′ − 2 Kℓ+ ′ Aℓ− ′ Rℓ− ′ Kℓ+ ′ ↔ − Σb (x, x ) = ig G0 (x, x )G0 (x , x) + G0 (x, x )G0 (x , x) + (+ ) . (A.58) ℓ

Wigner transforming this gives

R 2Im[Σb (x, q)] ∑ ∑ [ ( )( ) − 2 Rℓ+ − Aℓ+ Aℓ− − Rℓ− = g Ff (x, k + q) G0 (x, k + q) G0 (x, k + q) G0 (x, k) G0 (x, k) ℓ k ( )( ) ] Rℓ+ − Aℓ+ Rℓ− − Aℓ− ↔ − + Ff (x, k) G0 (x, k) G0 (x, k) G0 (x, k + q) G0 (x, k + q) + (+ ) ∫ ∑ 2 [ ( ) d k − = −g2 δ e (k) + ω − e+(k + q) F ℓ+(t, k + q)− 2π ℓ q ℓ f ℓ ( ) ] − + − − ℓ+ ↔ − δ eℓ (k) + ωq eℓ (k + q) Ff (t, k) + (+ )

R = 2Im[Σb (t, q, ωq)], (A.59)

245 where we have used spatial translational invariance and also have kept the external boson on shell. For the

Keldysh component of the boson self energy, the second diagram in Fig. 1.5 gives

K(2) iΣb (x, q) ∑ ∑ [ ( ) 2 Rℓ+ − Aℓ+ = g Ff (x, k + q) G0 (x, k + q) G0 (x, k + q) Ff (x, k) ℓ k ( ) ] × Rℓ− − Aℓ− ↔ − G0 (x, k) G0 (x, k) + (+ ) ∫ ∑ 2 [ ( ) ] d k − − = −ig2 δ ω + e (k) − e+(k + q) F ℓ+(t, k + q)F ℓ (t, k) + (+ ↔ −) 2π q ℓ ℓ f f ℓ

K(2) = 2iΣb (t, q, ωq). (A.60)

The first and third diagrams for the Keldysh component when combined give

K(1+3) iΣb (x, q) ∑ ∑ [( ) ( ) ] 2 Rℓ+ Aℓ+ Aℓ+ Rℓ− ↔ − = g G0 (x, k + q)G0 (x, k) + G0 (x, k + q)G0 (x, k) + (+ ) ℓ k ∫ ∫ [ ] ∑ 2 2 d k dω 1 1 = g − + c.c. + (+ ↔ −) 4π2 2π ω + ω − e+(k + q) + i0+ ω − e (k) − i0+ ℓ q ℓ ℓ ∫ ∑ 2 [ ( ) ] d k − = g2 δ ω + e (k) − e+(k + q) + (+ ↔ −) 2π q ℓ ℓ ℓ

K(1+3) = 2iΣb (t, q, ωq). (A.61)

Thus we obtain the boson collision integral:

∫ ∑ 2 [ ( ) d k − Icoll[F ,F ](t, q) = −g2 δ e (k) + ω − e+(k + q) b f b 2π ℓ q ℓ ℓ ( ) ] − − ℓ− ℓ+ ℓ− ℓ+ ↔ − 1 Ff (t, k)Fb(t, q, ωq) + Ff (t, k + q)Ff (t, k) + Ff (t, k + q)Fb(t, q, ωq) + (+ ) ,

(A.62)

246 or

∫ ∑ 2 [ ( ) ( d k − − Icoll[f , f ](t, q) = 4g2 δ e (k) + ω − e+(k + q) f ℓ+(t, k + q)(1 − f ℓ (t, k))+ b f b 2π ℓ q ℓ f f ℓ ) ] ℓ+ − ℓ− ↔ − + ff (t, k + q)fb(t, q, ωq) ff (t, k)fb(t, q, ωq) + (+ ) . (A.63)

A.4.4 Solution of the boson kinetic equation

Inserting the parametrization Eq. (1.65) for the fermion f functions into the boson collision integral and also parameterizing fb in the frequency domain as

fb(ω, q, ωq) = 2πδ(ω)nb(ωq) + u(ω, q, ωq), (A.64)

ℓ − − · ℓ + + · where u is linear in E. Changing variables to p1 = eℓ (k) = vℓ k and p2 = eℓ (k) = vℓ k, the boson collision integral (1.63) becomes

coll Ib [Ff ,Fb](t, q) ∫ 2 ∑ [ ( )( ( ) g − = dpℓ dpℓ δ pℓ + ω − pℓ − v+ · q f +(t, k + q) 1 − f (t, k) + πv 1 2 1 q 2 ℓ ℓ ℓ ℓ ) ] + − − ↔ − ↔ + fℓ (t, k + q)fb(t, q, ωq) fℓ (t, k)fb(t, q, ωq) + (+ , 1 2) . (A.65)

247 ℓ Always integrating out p2 in this expression, and keeping only terms up to linear order in E, we get, in the frequency domain, using the boson kinetic equation Eq. (1.60)

− + coll 2( iω + 0 )u(ω, q, ωq) = Ib [Ff ,Fb](ω) ∫ 2 ∑ [ g − = dpℓ a(pℓ , ω , v · q)(2πδ(ω)n (ω ) + u(ω, q, ω ))+ πv 1 1 q ℓ b q q ℓ

· ℓ ℓ − · · ℓ ℓ − · − · ℓ ℓ − · − + E(ω) b (p1, ωq, vℓ q)nb(ωq) + E(ω) b1(p1, ωq, vℓ q) E(ω) d (p1, ωq, vℓ q) ] − ℓ − · ℓ − · c(p1, ωq, vℓ q)(2πδ(ω)) + a1(p1, ωq, vℓ q)(2πδ(ω)) , (A.66)

248 where

ℓ − · ℓ − ℓ ℓ − · − ℓ − · − a(p1, ωq, vℓ q) = (nf (p1 + ωq) nf (p1)) + (nf (p1 + vℓ q) nf (p1 + vℓ q ωq)), ∫ ℓ ℓ − · − dp1a(p1, ωq, vℓ q) = 2ωq,

ℓ ℓ − · + ℓ − ℓ ℓ − − ℓ − ℓ ℓ b (p1, ωq, vℓ q) = vℓ nf (p1 + ωq)(1 nf (p1 + ωq))φ(p1 + ωq) vℓ nf (p1)(1 nf (p1))φ(p1)+

− ℓ − · − ℓ − · ℓ − · − + vℓ nf (p1 + vℓ q)(1 nf (p1 + vℓ q))φ(p1 + vℓ q)

− + ℓ − · − − ℓ − · − ℓ − · − vℓ nf (p1 + vℓ q ωq)(1 nf (p1 + vℓ q ωq))φ(p1 + vℓ q ωq),

ℓ ℓ − · + ℓ − ℓ ℓ b1(p1, ωq, vℓ q) = vℓ nf (p1 + ωq)(1 nf (p1 + ωq))φ(p1 + ωq)+

− ℓ − · − ℓ − · ℓ − · + vℓ nf (p1 + vℓ q)(1 nf (p1 + vℓ q))φ(p1 + vℓ q),

ℓ ℓ − · − ℓ ℓ − ℓ ℓ d (p1, ωq, vℓ q) = vℓ nf (p1 + ωq)nf (p1)(1 nf (p1))φ(p1)+

+ ℓ ℓ − ℓ ℓ + vℓ nf (p1)nf (p1 + ωq)(1 nf (p1 + ωq))φ(p1 + ωq)

+ ℓ − · ℓ − · − − ℓ − · − ℓ − · − + vℓ nf (p1 + vℓ q)nf (p1 + vℓ q ωq)(1 nf (p1 + vℓ q ωq))φ(p1 + vℓ q ωq)+

− ℓ − · − ℓ − · − ℓ − · ℓ − · + vℓ nf (p1 + vℓ q ωq)nf (p1 + vℓ q)(1 nf (p1 + vℓ q))φ(p1 + vℓ q),

ℓ − · ℓ ℓ ℓ − · ℓ − · − c(p1, ωq, vℓ q) = nf (p1 + ωq)nf (p1) + nf (p1 + vℓ q)nf (p1 + vℓ q ωq),

ℓ − · ℓ ℓ − · a1(p1, ωq, vℓ q) = nf (p1 + ωq) + nf (p1 + vℓ q), ∫ [ ] ∫ ℓ ℓ − · − ℓ − · ℓ ℓ − ℓ dp1 a1(p1, ωq, vℓ q) c(p1, ωq, vℓ q) = 2 dp1nf (p1 + ωq)(1 nf (p1)) = 2ωqnb(ωq).

(A.67)

ℓ ℓ ℓ ℓ − · Since each term in the b , b1, d terms results in a convergent integral over p1, the vℓ q s can be shifted ∑ ℓ ℓ ℓ out. Then, since ℓ vℓ = 0, the contribution from the b , b1 and d terms vanishes. Then,

g2 2(−iω + 0+)u(ω, q, ω ) = −8ω u(ω, q, ω ). (A.68) q q πv q

249 Since this has to hold for all values of ω, we can only have u = 0. Hence, the boson collision integral is trivially solved by the thermal Bose distribution and the bosons do not respond to the applied electric field in our approximation. It is also easily seen using the identity

nf (x)(1 − nf (x − y)) + nb(y)(nf (x) − nf (x − y)) = 0, (A.69)

that the thermal Fermi distribution nf nullifies the fermion collision integral in the absence of an applied electric field if the thermal Bose distribution nb is used for the bosons, as it should.

250 B Appendices to Chapter 2

B.1 Susceptibilities

The susceptibility χJP is taken to be the free fermion susceptibility at leading order and is thus given by

∫ ( ) 2 − d q 2 1 ∂nF (ξa(q)) 1 ∂nF (ξb(q)) χJP = 2 2 qx + . (B.1) (2π) 2m1 ∂ξa(q) 2m2 ∂ξb(q)

251 1/2 1/2 defining coordinates qx = (2m1,2) q1,2 cos θ, qy = (2m2,1) q1,2 sin θ, θ ∈ [0, 2π), q1,2 ∈ [0, ∞), so

2 2 that ξa = q1 and ξb = q2 the integral can be evaluated exactly to give

√ √ m1m2 µ0 m1m2 3 χ = T ln(1 + e T ) ≈ µ + O(T ) + ... , (B.2) JP π π 0

where µ0 ≫ T is the chemical potential for the fermions, and hence χJP is treated as a temperature- independent constant.

Both the Hamiltonian and P are invariant under SU(2) spin rotation, but the spin current transforms as a vector. Hence it may be easily seen that χSP = 0 since the contributions from states with opposite spins will cancel.

The linearized hot spot model has an emergent SU(2) particle-hole symmetry [84, 86], and one obtains

(with the hot spots indexed by l and the fermion types indexed by a)

1 ∑ J = v lΨl†σ Ψl , 2 a a z a l,a i ∑ P = (∇Ψl†Ψl − Ψl†∇Ψl ), (B.3) 4 a a a a l,a

( ) l l ψa l where Ψa = l† , ψa are two-component spinors, the τ matrices act only on the spin indices and the σ iτyψa matrices act only on the particle hole indices. The Lagrangian is invariant under the SU(2) transformations

l → l l iθl.σ/2 l l l Ψa U Ψa = e Ψa that rotate particles into holes. One can always choose U (for example U = iσx) such that J → −J and P → P , which implies that χJP = 0 in this case. If a curvature of the Fermi surface

l l · 2 l 2 l is introduced (the dispersion modified to ξa(q) = va q + qx/(2max) + qy/(2may)), this particle-hole symmetry is broken. We then have

[∫ ∫ ( ) ] ∑ Λ Λ − dqxdqy l qx ′ l χJP = 2 2 qx vax + l n (ξa(q)) . (B.4) − − (2π) 2m l,a Λ Λ ax

252 The particle-hole symmetric regularization is chosen to make the integral vanish when the quadratic terms from the dispersion are removed, as is required by the particle-hole symmetry in that case. The integral can

l now be expanded in 1/max,y to give

[ ] ∑ C (vl , Λ,T ) C (vl , Λ,T ) χ = 1 a + 2 a + ... , (B.5) JP ml ml l,a ax ay and hence the non-zero contributions are linear in the curvature to leading order. We emphasize here that this addition of a small curvature to the linear hot spot model is very different from the case of the two band model used throughout this chapter, which has curvature built in from the beginning, and hence does not have a small value of χJP that is perturbative in the curvature.

B.2 Computation of R(T )

Starting with our continuum model described by Eq. (2.1), we follow the Hertz strategy and integrate out the fermions to one loop order: As usual, only the coupling to the fermions near the hot spots modifies the boson propagator. We then consider the vector boson to have N components instead of 3 for the purpose of this computation, and subsequently take a large N limit. The effective Hertz action for the boson field then is

∫ ∫ ( ) d2q ∑ 1 [ ] u N 2 S = ϕ (q, ω )(q2 + γ|ω | + ϵω2)ϕ (−q, −ω ) + d2x dτ ϕ ϕ − . B (2π)2 2 µ q q q µ q 2N µ µ g ωq (B.6)

253 Decoupling the quartic interaction using an auxiliary field η gives

∫ d2q ∑ 1 [ ] S = ϕ (q, ω )(q2 + γ|ω | + ϵω2)ϕ (−q, −ω ) B (2π)2 2 µ q q q µ q ωq ∫ [ ( ) ] 2 2 1 iη N η + d x dτ √ ϕµϕµ − + . (B.7) 2 N g 4u

We now take u → ∞, making the above action equivalent to that for an O(N) non-linear sigma model with a fixed length constraint. Considering η to be constant, we integrate out ϕµ to obtain the one loop (equivalently

N = ∞) effective potential density for η:

√ ∫ ( ) 2 ∑ iη N N d q 2 2 iη Veff = − T ln q + γ|ωq| + ϵω + √ , (B.8) 2g 2 (2π)2 q N ωq

√ using iη/ N = R(T ) and R(0) = 0 at the critical point g = gc and minimizing this yields (while approach- ing the critical point from the g > gc side)

∫ d2q ∑ 1 1 2 T 2 2 = + , (2π) q + γ|ωq| + ϵω + R(T ) gc + 0 ωq q ∫ ∫ ∫ 2 ∑ 2 d q 1 − d q dωq 1 2 T 2 2 2 2 2 = 0. (B.9) (2π) q + γ|ωq| + ϵω + R(T ) (2π) 2π q + γ|ωq| + ϵω + δ+ ωq q q

Where δ+ is a small positive regulator. We subtract the following from the first term in the last line of the above (and add it to the second term):

∫ ∫ d2q dω 1 q ; (B.10) 2 2 | | 2 (2π) 2π q + γ ωq + ϵωq + R(T )

254 The frequency summation in the first term is carried out by analytically continuing |ω| using the following identities ∫ ∫ iω ∞ dx i ∞ dx |ω| = − ; sgn(ω) = − , (B.11) π −∞ x − iω π −∞ x − iω which gives 1 1 → , (B.12) 2 | | 2 2 − 2 − q + γ ωq + ϵωq + R(T ) q ϵz iγzsgn(Im[z]) + R(T ) and avoiding the discontinuity along the real axis in the contour integration over z (The function has no poles as R(T ), γ, ϵ > 0). We obtain

∫ ∑ 1 ∞ dω γω T = 2 n (ω) + 2 | | 2 2 − 2 2 2 2 B q + γ ωq + ϵωq + R(T ) 0 π (q ϵω + R(T )) + γ ω ∫ ωq ∞ dω γω 2 2 2 2 2 . (B.13) 0 π (q − ϵω + R(T )) + γ ω

The limit δ+ → 0 can be taken at the end without any disastrous consequences. Finally, we obtain:

∫ [ ( )] ( ) ∞ − 2 π − −1 R(T ) ϵω R(T )ϵ 4ϵ dω tan nB(ω) + γ ln 2 0 2 ( (γω ) γ ) √ γ ( ) + 4R(T )ϵ − γ2 2 tan−1 √ − πsgn 4R(T )ϵ − γ2 = 0. (B.14) 4R(T )ϵ − γ2

This may be solved numerically for R(T ), however one finds that (See Fig. B.1), to a good approximation,

R(T ) is described by the simple form γT + ϵT 2 at intermediate values of T . In the z = 2 limit (ϵ → 0), we have (Λ is a UV momentum cutoff required as a regulator in this limit)

( ) ∫ ( ) 2 ∞ Λ −1 γω R(T ) ln = 2γ dω tan nB(ω), (B.15) R(T ) 0 R(T )

255 which gives ( ) γT R(T ) = γT f , (B.16) Λ2 where f is a very slowly varying function with the property f(0) = 0. We find

( ( )) − 1 eγE 1 πW0 ln ≈ π( 2)πx f(x) − , (B.17) eγE 1 ln 2πx

where W0 is the principal branch of the Lambert W function, and γE is the Euler-Mascheroni constant. In the opposite limit of z = 1 (γ → 0), we get the exact result [348]

[ ( )] √ 2 5 + 1 R(T ) = ϵT 2 2 ln . (B.18) 2

Figure B.1: Numerical solution (solid) of Eq. B.14, and γT + ϵT 2 (dashed), for ϵ = 1 and γ = 1.

256 B.3 Random Mass Computations

We construct expressions for Π(˜ k, iΩ) in terms of the spectral function for the vector boson Green’s function:

−2γEδ A (q,E) = µν . (B.19) µν (q2 − ϵE2 + R(T ))2 + γ2E2

We have

∫ ∑ 2 − ˜ d q dE1 dE2 Aµν(q,E1) Aνµ(q k,E2) Π(k, iΩ) = T 4 , (2π) iωq − E1 iωq − iΩ − E2 ωq ∫ Im[Π˜ R(k, ω)] d2q dE −γ2E2 lim = 6 1 1 → 3 2 − 2 2 2 2 − 2 − 2 2 2 2 ω 0 ω (2π) [(q + R(T ) ϵE1 ) + γ E1 ][((q k) + R(T ) ϵE1 ) + γ E1 ] × ′ nB(E1), ∫ d2q dE dE lim Re[Π˜ R(k, ω)] = 12 1 2 ω→0 (2π)4 γ2E E n (E ) − n (E ) × 1 2 B 2 B 1 . (B.20) 2 − 2 2 2 2 − 2 − 2 2 2 2 − [(q + R(T ) ϵE1 ) + γ E1 ][((q k) + R(T ) ϵE2 ) + γ E2 ] E1 E2

In the z = 2 limit, performing the frequency integrals gives

∫ [ ( ) Im[Π˜ R(k, ω)] 3 d2q 1 q2 + R(T ) q2 + R(T ) lim = ψ′ ω→0 ω 2πγT (2π)2 ((q − k)2 + R(T ))2 − (q2 + R(T ))2 2π 2πγT ( ) ( ) ] (q − k)2 + R(T ) (q − k)2 + R(T ) 1 1 − ψ′ + πγ2T 2 − , 2π 2πγT (q − k)2 + R(T ) q2 + R(T ) ∫ [ ( ) ( ) 6 Λ d2q 1 q2 + R(T ) (q − k)2 + R(T ) lim Re[Π˜ R(k, ω)] = ψ − ψ + ω→0 2πγ (2π)2 q2 − (q − k)2 2πγT 2πγT ( ) ] 1 1 πγT − , (B.21) q2 + R(T ) (q − k)2 + R(T )

257 where ψ here is the digamma function, and Λ is a momentum cutoff. We obtain the following asymptotic forms in k (only the dependence on k, T, Λ and γ is shown)

( ) Im[Π˜ R(k, ω)] γT γT lim ∼ ln f , k2 ≫ γT, ω→0 ω k4 Λ2 [ ( )]− Im[Π˜ R(k, ω)] 1 γT 2 lim ∼ f , k2 ≪ γT, ω→0 ω γT Λ2 1 lim Re[Π˜ R(k, ω)] ∼ , k2 → Λ2 ≫ γT, ω→0 γ ( [ ( )] ( )) −1 2 R 1 γT Λ 2 lim Re[Π˜ (k, ω)] ∼ b1 f + b2 ln , k ≪ γT, (B.22) ω→0 γ Λ2 γT where f is the function correcting the linear dependence of R(T ) on T defined in Eq. (B.17). We then have

∫ √ ∫ 2 γT Λ ˜ R m0 1 3 Im[Π (k, ω)] ρxx(T ) ∝ lim + k dk . (B.23) 2 2 → √ R 2 u |χJP | ω 0 ω 0 γT Re[Π˜ (k, ω)]

Substituting the small k asymptotic forms in the first integral and the large k ones in the second, and noting that f(x) ∼ ln ln(1/x)/ ln(1/x), we obtain the scaling form given in the main text to leading-log order, which agrees well with numerical evaluation of the integrals.

For the z = 1 limit, we have,

 (√ √ ) ∫ − ′ 2 2 Im[Π˜ R(k, ω)] 3 π/2 dθ nB k /(4 cos θ) + R(T )/ ϵ lim = √  √  , → 2 2 2 ω 0 ω 16π ϵ −π/2 cos θ k /(4 cos θ) + R(T ) [ (√ /√ ) ∫ − 2 3 d2q 1 2nB (q k) + R(T ) ϵ + 1 lim Re[Π˜ R(k, ω)] = √ √ − ω→0 2 ϵ (2π)2 q2 − (q − k)2 (q − k)2 + R(T ) (√ √ ) ] 2 2nB q + R(T )/ ϵ + 1 √ . (B.24) q2 + R(T )

These integrals are convergent, and we can thus scale out ϵT 2 after plugging in R(T ) to get the result in the main text.

258 In the crossover region between the z = 2 to z = 1 regime, we evaluate all integrals numerically and plug in the numerical solution for R(T ) at arbitrary T to obtain Fig. 2.3.

B.4 Vertex Correction for Inter Hot-Spot Scattering

We now compute the graph in Fig. 2.4(b), which is the leading vertex correction to the resistivity for inter hot-spot scattering. In the approximation of Eq. 2.26, the momenta flowing through the upper and lower fermion lines in the graph are independent of each other. Since the bare fermion propagator depends only on the component of its momentum transverse to the fermi surface, and because the interaction with the boson switches the fermion type, we have (using the spectral representation for the boson Green’s function):

[ ∫ 2 2 ∑ ∑ ij2 2 ∑ 6V0 λ Qx dξiαdξiα¯dξjβdξjβ¯dEd q 2 1 ρxx(T ) = − lim Im T × ω→0 | |2 | × || × | 7 − ω χJP ̸ viα viα¯ vjβ vjβ¯ (2π) iωq ξiα i,j,i=j α,β ωq ,η ] 1 1 1 1 −2γE − − · − − − · − − − 2 2 2 2 iωq + iη ξiα¯ viα¯ q iωq + iη iΩ ξjβ vjβ q iωq iΩ ξjβ¯ iη E (q + R(T )) + γ E iΩ→ω+i0+ [ ∫ 2 2 ∑ ∑ ij2 ∑ 6V λ Q dξiαdξiα¯dξjβdξ ¯dE 1 = − lim 0 x Im jβ T 2 × → | |2 | × || × | 6 − ω 0 ω χJP viα viα¯ vjβ vjβ¯ (2π) iωq ξiα i,j,i̸=j α,β ωq ,η ( ( ) ) ] 1 1 1 1 −1 R(T ) − π − − − − − − tan , iωq + iη ξiα¯ iωq + iη iΩ ξjβ iωq iΩ ξjβ¯ iη E γE 2 iΩ→ω+i0+

(B.25)

¯ where the indices α, β run over the fermion types a, b, a¯ = b, b = a, ωq is a fermionic Matsubara frequency, and η, Ω are bosonic Matsubara frequencies. In the second step in the above, we have used the independence of the ξiα’s to shift out the boson momenta entering the fermion propagators. One should note that here since all the fermion propagators have independent ξiα’s, factors ∫ of dξ/(iω − ξ)m≥2 = 0 do not appear even when the boson momentum and frequency go to zero, and the most singular contribution of the graph thus survives. This will not be the case for the higher order corrections mentioned at the end of this appendix. After carrying out the frequency

259 summations, We have to evaluate

∫ ( dξiαdξiα¯ dξjβ dξ ¯dE n (E) n (ξ ) Im jβ − B F iα( ) − 6 (2π) (ξiα − ξiα¯ + E)(−iΩ + ξiα − ξiα¯ ) −iΩ + ξiα − ξjβ + E ( ) ( ) n (E) n ξ ¯ nB (E) nF ξjβ − E B F jβ ( ) ( )( ) − ( )( )( ) − − − − − − − − − ξjβ ξjβ¯ E iΩ ξiα¯ + ξjβ iΩ ξiα + ξjβ E ξjβ + ξjβ¯ + E iΩ ξiα + ξjβ¯ iΩ ξiα¯ + ξjβ¯ + E

nB (E) nF (ξiα¯ − E) nF (ξiα¯ ) nF (ξiα) ( ) ( ) + ( ) ( ) − − − − − − − − − − − − − − ( ξiα + ξiα¯ E) iΩ + ξiα¯ ξjβ iΩ + ξiα¯ ξjβ¯ E ( ξiα + ξiα¯ E) iΩ + ξiα ξjβ¯ iΩ + ξiα¯ ξjβ ( ) ( ) nF (ξiα¯ ) nF (ξiα¯ − E) nF ξjβ nF ξjβ − E ( ) ( ) − ( ) ( )( ) + − − − − − − − − − − − − ( ξiα + ξiα¯ E) iΩ + ξiα¯ ξjβ iΩ + ξiα¯ ξjβ¯ E ξjβ ξjβ¯ E iΩ ξiα + ξjβ E iΩ ξiα¯ + ξjβ ( ) ( ) ( ) nF ξjβ nF ξjβ¯ nF (ξiα¯ ) nF ξjβ¯ ( )( ) ( ) + ( )( ) ( ) − − − − − − − − − − ξjβ ξjβ¯ E iΩ ξiα + ξjβ¯ iΩ ξiα¯ + ξjβ iΩ ξiα + ξjβ¯ iΩ + ξiα¯ ξjβ¯ E iΩ + ξiα¯ ξjβ ) ( ) ( ( ) ) nF ξjβ nF (ξiα) − R(T ) π + ( )( ) ( ) × tan 1 − , − − − − − γE 2 iΩ ξiα¯ + ξjβ iΩ ξiα + ξjβ E iΩ + ξiα ξjβ¯ iΩ→ω+i0+

(B.26)

Using 1/(x + i0) = ∓iπδ(x) + P/x, the imaginary parts of the first eight terms inside the

↔ ↔ brackets in the above vanish. For the last term, relabeling dummy variables ξiα ξiα¯, ξjβ ξjβ¯

and E → −E simplifies the above expression to

( ) ∫ ( ) 2n (ξ ) n ξ ¯ dξiαdξiα¯ dξjβ dξjβ¯dE F iα¯ F jβ − R(T ) Im ( )( ) ( ) tan 1 (2π)6 − − − − − − γE iΩ ξiα + ξjβ¯ iΩ + ξiα¯ ξjβ¯ E iΩ + ξiα¯ ξjβ iΩ→Ω+i0+ ∫ ( ) dξ dξ dξ dξ dE ( ) 2 iα iα¯ jβ jβ¯ −1 R(T ) = π nF (ξiα¯ ) nF ξ ¯ δ(ω − ξiα + ξ ¯)δ(−ω + ξiα¯ − ξ ¯ − E)δ(−ω + ξiα¯ − ξ ) tan (2π)5 jβ jβ jβ jβ γE ( ) ∫ ( ) 1 − R(T ) = dξ dξ n (ξ ) n ξ tan 1 . (B.27) 3 iα¯ jβ¯ F iα¯ F jβ¯ − − 32π γ(ξiα¯ ξjβ¯ ω)

↔ If ω = 0 this evaluates to 0 as the integrand is odd under ξiα¯ ξjβ¯. Hence we have

∫ 2 2 ∑ ∑ Λ˜ − 3V0 λ ij2 dξiα¯dξjα¯ γR(T ) ρxx(T ) = 3 2 Qx nF (ξiα¯) nF (ξjα¯) 2 2 2 , 16π |χJP | −˜ |viα × viα¯||vjβ × v ¯| γ (ξiα¯ − ξjα¯) + R(T ) i,j,i≠ j α,β Λ jβ (B.28)

260 ˜ Where the cutoff Λ ≫ T is used to regulate the divergence of the integral as ξiα, ξiα¯ → −∞. We decompose the integration into four quadrants and obtain to leading-log order in T :

∫ ∫ [ ( )] Λ˜ Λ˜ γR(T ) ≈ γT I++ = dξiα¯dξjα¯nF (ξiα¯) nF (ξjα¯) 2 2 2 T a1 + a2f 2 , γ (ξiα¯ − ξjα¯) + R(T ) Λ 0 0 ∫ ∫ ( ) Λ˜ 0 γR(T ) γT − − ≈ I+ = I + = dξiα¯dξjα¯nF (ξiα¯) nF (ξjα¯) 2 2 2 a3T f 2 , −˜ γ (ξiα¯ − ξjα¯) + R(T ) Λ ∫ ∫ 0 Λ [ ( )] 0 0 reg − γR(T ) ≈ γT I−− = dξiα¯dξjα¯(nF (ξiα¯) nF (ξjα¯) 1) 2 2 2 T a4 + a5f 2 , −˜ −˜ γ (ξiα¯ − ξjα¯) + R(T ) Λ ∫ Λ ∫ Λ ( ) 0 0 γR(T ) R(T ) R(T ) Idiv = dξ dξ ≈ πΛ˜ + 2 ln , (B.29) −− iα¯ jα¯ 2 − 2 2 −Λ˜ −Λ˜ γ (ξiα¯ ξjα¯) + R(T ) γ γ˜Λ

√ √ where a1 = π ln(2/ e), a4 = −π ln(2 e), and a2,3,5 have very slow log-log dependences on T .

Here f is the function correcting the linear dependence of R(T ) on T and is defined in Eq. B.17,

and Λ is the cutoff used in Eq. B.15. We thus obtain

[ ] V 2λ2 ∑ ∑ 1 T ≈ 0 ij2 − ˜ ρxx(T ) 2 Qx aΛ + bT + c 2 , (B.30) |χJP | |viα × viα¯||vjβ × v ¯| ln(Λ /(γT )) i,j,i≠ j α,β jβ

at low T where a, b > 0 and b, c have very slow log-log dependences on T .

In the z = 1 limit, the factor of tan−1(R(T )/(γE))−π/2 in Eq. B.25 is replaced with −πΘ(ϵ(E2−

2 ′ ˜ ′ ′ ′ cT )). Then, performing the same computation yields ρxx(T ) ∼ −a Λ + b T , a , b > 0.

We can also consider other graphs which have a fermion loop that runs through both the external vertices, and multiple internal boson propagators that intersect this loop at various points (For example, one such family of graphs would be the higher order graphs in the “ladder series” of graphs, which contain multiple boson propagators connecting the upper and lower fermion lines instead of just one in the above vertex correction). The most singular contribution from these graphs

261 would arise when the momenta and frequencies of all these internal boson propagators go to zero

simultaneously: When this happens, such graphs will be given by expressions of the form

∫ ∑ ∑ ij2 ∑ Qx 1 1 ¯ T dξiαdξiα¯dξjβdξjβ t t |viα × viα¯||vjβ × v ¯| (iωq − ξiα) 1 (iωq − ξiα¯) 2 i,j,i≠ j α,β jβ ω q ( ) 1 1 T n × , (B.31) − − t3 − − t4 (iωq iΩ ξjβ) (iωq iΩ ξjβ¯) R(T ) where n is the number of internal boson propagators. It is guaranteed that at least one of the t’s is

≥ 2, because at least one of the fermion lines will have more than one intersection with an internal boson propagator if there is more than one internal boson propagator. Hence this expression to evaluates to zero, and the most singular contribution vanishes.

262 C Appendices to Chapter 3

C.1 Solution to linearized disordered magneto-hydrodynamic equa-

tions

At O(ϵ) we get the following equations in momentum space from (3.4)

( ) − (1) (0) (1) (0) (1) − (0) (1) (0) 2 (1) (1) Q δEi + iki Q δµ + S δT iζ kjvj + η k vi = BϵijJj , (C.1) δE J (1) = Q(0)v(1) + Q(1)ϵ j − σQ(0)(ik δµ(1) − Bϵ v(1)) − iαQ(0)k δT (1), i i ij B ij j jk k ij j

263 δE J H(1) = TS(0)v(1) + TS(1)ϵ j − T α¯Q(0)(ik δµ(1) − Bϵ v(1)) − iκ¯Q(0)k δT (1), (C.2) i i ij B ij j jk k ij j

(1) H(1) kiJi = kiJi = 0. (C.3)

(1) (1) The zero-momentum limit of the equations (C.3) implies that vi and Ji have only finite momen-

tum components, since the periodic boundary conditions ensure that limk→0 kiδµ = limk→0 kiδT =

0. Q(1) also has only finite momentum components and a solution to

(0) (1) − Q(0) (1) Q vi (k = 0) = Bσij ϵjkvk (k = 0) (C.4)

doesn’t exist. Hence we need to look at O(ϵ2) to obtain nontrivial uniform conductivities.

(1) Using the first equation of (C.3) to read off Ji , we obtain

ϵ k J (1) = − ij k (η(0)k2v(1) − Q(1)δE ). (C.5) i i B i j j

Note that it is impossible to set the right hand side of this equation to zero in the limit of vanishing

shear viscosity η(0). Thus, this perturbative expansion is valid only for finite background shear viscosities.

264 The solution to (C.3) is given by

( ( ) (1) 4 (1) Q(0) Q(0) Q(0) Q(0)2 δµ = iϵijEikj B Q σ κ¯ σ − α T

( ( ) + B2 αQ(0)Q(1)T −αQ(0)k2(η(0) + ζ(0)) − 2Q(0)S(0) ( ) ) + σQ(0) (ζ(0) + 2η(0))¯κQ(0)k2Q(1) + αQ(0)η(0)k2(−S(1))T + Q(1)S(0)2T +κ ¯Q(0)Q(0)2Q(1) ) ( ( ) ) + η(0)k2 S(1)T −αQ(0)k2(η(0) + ζ(0)) − Q(0)S(0) +κ ¯Q(0)k2Q(1)(ζ(0) + η(0)) + Q(1)S(0)2T ( ( ( ) × Bη(0)k4 B2κ¯Q(0)σQ(0)2 + σQ(0) T (αQ(0)B + S(0))(S(0) − αQ(0)B) +κ ¯Q(0)k2(ζ(0) + η(0)) ) ) −1 − αQ(0)2k2T (ζ(0) + η(0)) +κ ¯Q(0)Q(0)2 − 2αQ(0)Q(0)S(0)T , (C.6)

δT (1) = ( ) ( ) Q(0) (1) (1) Q(0) 2 Q(0) 2 (0) (0) (0)2 (1) (0) (1) (0) − iϵijEikjT (α Q − S σ ) B σ + k (ζ + η ) − Q S + Q Q S ( ( ( ) × Bk2 B2κ¯Q(0)σQ(0)2 + σQ(0) T (αQ(0)B + S(0))(S(0) − αQ(0)B) +κ ¯Q(0)k2(ζ(0) + η(0)) ) ) −1 − αQ(0)2k2T (ζ(0) + η(0)) +κ ¯Q(0)Q(0)2 − 2αQ(0)Q(0)S(0)T , (C.7)

(1) vi ( ( ) ϵ E k ϵ k Q(1) = lm l m ij j − k2k (−κ¯Q(0)Q(0)Q(1) + αQ(0)T (Q(0)S(1) + Q(1)S(0)) − S(0)S(1)σQ(0)T ) k4 η(0) i ( ( ( ) × B B2κ¯Q(0)σQ(0)2 + σQ(0) T (αQ(0)B + S(0))(S(0) − αQ(0)B) +κ ¯Q(0)k2(ζ(0) + η(0)) ) )) −1 − αQ(0)2k2T (ζ(0) + η(0)) +κ ¯Q(0)Q(0)2 − 2αQ(0)Q(0)S(0)T . (C.8)

265 Using (3.5), the expression for the uniform electrical conductivity σij retaining only the leading

and next-to-leading contributions in the inverse disorder wavelength in the diagonal conductivity

is given by

∫ (0) 2 (2) 2 | (1) |2 (0) 2 (2) Q + ϵ Q (k = 0) ϵ klkm Q (k) σij = σij + ϵ σij = ϵij + (0) ϵilϵjm 4 B η k k ϵ2 + η(0)B2 (σQ(0) (B2 (¯κQ(0)σQ(0) − αQ(0)2T ) + S(0)2T ) +κ ¯Q(0)Q(0)2 − 2αQ(0)Q(0)S(0)T ) ∫ [ k k 2 × l m (0) (1) − (0) − (1) (0) ϵilϵjm 2 η T Q ( k)S S (k)Q k k ( − B2Q(1)(k) − κ¯Q(0)σQ(0)(ζ(0) + 2η(0))Q(1)(−k) + αQ(0)2T (ζ(0) + η(0))Q(1)(−k) ) ] + αQ(0)η(0)σQ(0)TS(1)(−k) + B2η(0)(−T )S(1)(k)(σQ(0)(αQ(0)Q(1)(−k) − σQ(0)S(1)(−k))) ( ) ϵ2 (ζ(0) + η(0)) αQ(0)2T − κ¯Q(0)σQ(0) + η(0) σQ(0) (B2 (¯κQ(0)σQ(0) − αQ(0)2T ) + S(0)2T ) +κ ¯Q(0)Q(0)2 − 2αQ(0)Q(0)S(0)T ∫ | (1) |2 × klkm Q (k) ϵilϵjm 2 . (C.9) k k

Note that the leading disorder-induced contribution depends only upon the shear viscosity η(0), and that all corrections coming from the microscopic incoherent conductivities occur at higher orders in the inverse disorder wavelength.

In the presence of a magnetic field, the stress tensor Tij can contain the effects of new parity- odd microscopic transport coefficients. These are the Hall viscosity ηH [169] and the vorticity

susceptibility χΩ [170], which are both proportional to B. The stress tensor is modified to

η T → T + H (ϵ (∂ v + ∂ v ) + ϵ (∂ v + ∂ v )) + χ δ ϵ ∂ v . (C.10) ij ij 2 ik k j j k jk k i i k Ω ij kl k l

266 At O(ϵ), this becomes

η(0) T (1) → T (1) + i H (ϵ (k v(1) + k v(1)) + ϵ (k v(1) + k v(1))) + iχ(0)δ ϵ k v(1). (C.11) ij ij 2 il l j j l jl l i i l Ω ij lm l m

Then, repeating our solution, we find that the long-wavelength disorder result for the magnetore-

sistance given by (3.7) is unaffected by these terms.

C.2 Magneto-thermal transport in the clean system

To obtain the thermal resistance,

κxx + κyy Tr ρth = , (C.12) κxxκyy − κxyκyx

(0) (0) (0) we apply a temperature gradient ∂iT = −ξxxˆ, an electric field Ex = −S ξx/Q to block

electric currents, and solve the hydrodynamic equations in linear response. The clean solution is

S(0)σQ(0) − αQ(0)Q(0) v(0) = ξ , (C.13) x x B2σQ(0)2 + Q(0)2 BσQ(0) v(0) = v(0). (C.14) y Q(0) x

This choice gives J = 0 but J H ≠ 0. Therefore, there is a finite magneto-thermal resistance in the

clean system itself, given by

( ( )) 2 (0)2 Q(0) (0) − (0) Q(0) 2 Q(0)2 − Q(0) Q(0) 2B Q T (α Q S σ ) α T κ¯ σ 4 Tr ρth(B) − Tr ρth(0) = + O(B ). (2αQ(0)Q(0)S(0)T − S(0)2σQ(0)T − κ¯Q(0)Q(0)2)3

(C.15)

267 Unlike the charge magneto-transport in the clean system, the magneto-thermal resistance is ac-

Q(0) tually sensitive at the same order in B to the off-diagonal microscopic conductivities σxy =

Q(0) Q(0) Q(0) Q(0) Q(0) Q(0) Q(0) −σyx = a1B, κ¯xy = −κ¯yx = a3B and αxy =α ¯xy = −αyx = −α¯yx = a2B at small B.

Thus (C.15) should be modified to

2B2Q(0)2 Tr ρth(B) − Tr ρth(0) = Q(0) (0)2 Q(0) (0) (0) Q(0) (0)2 3 ( (¯κ Q + 2α Q S T + σ S T ) ( ) × 2 (0)4 − (0) (0)2 (0) − (0) (0)2 Q(0) (0) − Q(0) (0) 2 a3Q 2a3Q T 2a2Q S a1Q S + (α Q σ S )

( ( ) Q(0) (0) − Q(0) (0) Q(0) (0)2 − Q(0) (0)2 2 (0)2 (0)2 + T 2a2(α Q σ S ) κ¯ Q σ S T + 4a2Q S T

− (0) (0)3 − (0) Q(0) Q(0) − Q(0) (0) Q(0) (0) − Q(0) (0) 2 (0)4 4a1a2Q S T 2a1S (σ S α Q )(α S T κ¯ Q ) + a1S T ) ( ) ) + (αQ(0)Q(0) − σQ(0)S(0))2 αQ(0)2T − κ¯Q(0)σQ(0) + O(B4). (C.16)

C.3 Solution to hydrodynamic equations of the bilayer system

Since we have a uniform velocity field vx in layer 1, we modify its force equation from (3.1) to

d (M v1) + ∂ T 1 = 0, (C.17) dt 1 i j ij

1 where d/dt = ∂t + vj ∂j includes the contribution of converting from a co-moving frame. This is

a non-linear effect which is present in both Galilean and relativistic hydrodynamics, for which the

full non-linear theories are known.

After linearization in the noise sources and vx, the force equations for the two layers in momen-

268 tum/frequency space become

( ) 2π iM(v k − ω)δv1 − iωv δM + ik b δQ + b δS + Q(0)(δQ + e−kdδQ ) x x i x 1 i 1 1 2 1 k 1 2

(0) (0) 2 1 1 + (ζ kikj + η k δij)δvj + ikjsij = 0, ( ) 2π − iMωδv2 + ik b δQ + b δS + Q(0)(δQ + e−kdδQ ) i i 1 2 2 2 k 2 1

(0) (0) 2 2 2 + (ζ kikj + η k δij)δvj + ikjsij = 0, (C.18)

where

( ) ( ) ∂µ ∂T b = Q(0) + S(0) , (C.19) 1 ∂Q ∂Q ( ) S S (0) ∂µ (0) T b2 = Q + S . (C.20) ∂S Q CV

(0) At low temperatures we also expect δM1 ≈ (M/Q )δQ1 in non-Fermi liquids with a Fermi

surface (See (3.15)). The above equations then provide

(0) −kd 2 − (0) 1 1 2πkQ (δQ1 + e δQ2) + k (b1δQ1 + b2δS1) kxvxω(M/Q )δQ1 + kisijkj kiδvi = 2 , M(iDηk + ω − kxvx) (C.21)

(0) −kd 2 2 2 2πkQ (δQ2 + e δQ1) + k (b1δQ2 + b2δS2) + kisijkj kiδvi = 2 , (C.22) M(iDηk + ω)

(0) (0) Q(0) Q(0) (0) with Dη = (η + ζ )/M. To disable thermal currents, we set κ¯ = α = S = 0. This

269 makes δS1,2 = 0. Inserting (C.22) into the continuity equations for charge, we obtain

2 − 2 − (0) (0) −kd 2 1 M(Dσk iω)(Dηk iω)δQ1 + Q (2πkQ (δQ1 + e δQ2) + b1k δQ1 + kisijkj)

2 − − 2 − 1 + iM((Dσ + Dη)k iω)kxvxδQ1 = iM(Dηk iω + ikxvx)kiri , (C.23)

2 − 2 − (0) (0) −kd 2 1 M(Dσk iω)(Dηk iω)δQ2 + Q (2πkQ (δQ2 + e δQ1) + b1k δQ2 + kisijkj)

− 2 − 2 = iM(Dηk iω)kiri , (C.24)

Q(0) with Dσ = σ (∂µ/∂Q)S. In the basis, (C.24) turns into (3.12).

In the Halperin-Lee-Read (HLR) composite fermion theory of the ν = 1/2 quantum Hall state [31],

the composite fermions are also subjected to a transverse magnetic field corresponding to the devi-

ation in their local density from half-filling. Thus, to linear order in the noise terms, we should also

∝ (0) 1 ∝ Q(0) add a term δiyδQ1Q vx to the right hand side of (C.17) and shift Jy by a term σ δQ1vx.

However, these terms end up producing parity-odd contributions to ρD that vanish upon integration over ky, and can thus be ignored.

The frequency integrations in (3.14) may be performed first

∫ ∫ ∞ 4πQ(0)2k3e−kdM(D + D ) ∞ ω (ω2 − ω2 ) dω σ η = dω 0 + − | |2| |2 2 − 2 2 2 2 2 − 2 2 2 2 −∞ Π+ Π− −∞ ((ω ω+) + ω ω0)((ω ω−) + ω ω0) 2 − 2 2 2 2 π(ω+ ω−)(2ω0 + ω+ + ω−) = 2 2 2 2 2 2 2 2 . (C.25) ω ω−((ω − ω−) + 2ω (ω + ω−)) ∫ + + 0 + ∞ k2ω2M(D + D ) 2π/M dω σ η = , (C.26) | |2| |2 2 − 2 2 2 2 2 −∞ Π+ Π− (ω+ ω−) + 2ω0(ω+ + ω−)

2 3 (0) (0) ω = k(DσDηk M + Q (kb1 + 2πQ χ)), (C.27)

2 4 2 ω0 = Mk (Dσ + Dη) . (C.28)

270 We then get

∫ ∞ 2πT σQ(0)Mk3e−2kd ρσ = dk (C.29) D 16π2Q(0)4e−2kd + 4k3M(D + D )2(D D k3M + Q(0)(kb + 2πQ(0))) ∫ 0 η σ η σ 1 ∞ Q(0) 2 2 6 −2kd 3 2 3 (0) (0) πT σ D M k e (k M(Dη + Dσ) + DηDσk M + Q (kb1 + 2πQ )) − dk η 3 (0) (0) 2 − 2 (0)4 −2kd 0 (DηDσk M + Q (kb1 + 2πQ )) 4π Q e 1 × , 8π2Q(0)4e−2kd + 2k3M(D + D )2(D D k3M + Q(0)(kb + 2πQ(0))) ∫ η σ η σ 1 ∞ πT (η(0) + ζ(0))Q(0)2k4e−2kd(k3M(D + D )2 + D D k3M + Q(0)(kb + 2πQ(0))) ρη = dk η σ η σ 1 D 3 (0) (0) 2 − 2 (0)4 −2kd 0 (DηDσk M + Q (kb1 + 2πQ )) 4π Q e × 1 2 (0)4 −2kd 3 2 3 (0) (0) . (C.30) 8π Q e + 2k M(Dη + Dσ) (DηDσk M + Q (kb1 + 2πQ ))

These expressions can be written in the following scaling form, restoring factors of kB and ε,(dc =

2 (0)2 1/3 2 2 (εM(Dη + Dσ) /Q ) , de = ℏ ε/(e m))

∫ k T σQ(0)M ∞ p3e−2p ρσ = B dp (C.31) D 4d4Q(0)4 2πe−2p + p3(d /d)3(1 + pd /(2πd)) + ∆(p6/(2π))(d /d)6 0∫ c e c Q(0) 2 2 ∞ 6 −2p 3 3 kBT σ D M ε p e ((p /(2π))(d /d) (1 + ∆) + 1 + pd /(2πd)) − η dp c e 7 (0)6 3 3 2 − −2p 8πd Q 0 (∆(p /(2π))(dc/d) + 1 + pde/(2πd)) e 1 × , 2πe−2p + p3(d /d)3(1 + pd /(2πd)) + ∆(p6/(2π))(d /d)6 c ∫ e c k T (η(0) + ζ(0))ε ∞ p4e−2p((p3/(2π))(d /d)3(1 + ∆) + 1 + pd /(2πd)) ρη = B dp c e D 5 (0)4 3 3 2 − −2p 8πd Q 0 (∆(p /(2π))(dc/d) + 1 + pde/(2πd)) e × 1 −2p 3 3 6 6 . (C.32) 2πe + p (dc/d) (1 + pde/(2πd)) + ∆(p /(2π))(dc/d)

2 Where ∆ = DηDσ/(Dη + Dσ) is independent of d and T for the Fermi surface coupled to

U(1) gauge field. We can neglect the small de. At large d ≫ dc, we consider the first term in

σ ≫ ≲ ρD: The exponential in the numerator implies that at large d dc, the region of interest is p 1.

In this region, the first term of the denominator dominates the second. We can thus approximate

271 the integrand by p3/(2π). The integral is cut off when the two terms in the denominator become

comparable, i.e. when

( ( ) ) ( ) (0)2 3 1/3 ≈ 3 2 2πQ d ∼ d p W 2 ln , (C.33) 2 3 M(Dη + Dσ) dc where W (x) is the Lambert W function, which has the property W (x ≫ 1) ≈ ln x − ln ln x. The

η σ same strategy can be used to do the integral for ρD. The second term in ρD falls off much faster

faster with d than the first term. It is also non-singular as T → 0. We can thus ignore it. We then

obtain

( ) k T σQ(0)M d ρσ ∼ B ln4 , (C.34) D Q(0)4d4 d c ( ) (0) (0) η ∼ kBT (η + ζ )ε 5 d ρD (0)4 5 ln , (C.35) Q d dc

which translates to (3.17). At small separations d ≪ dc we can set the exponential factors in

(C.32) to unity, without making the integrals UV divergent. Then, all the contributions become

σ ∼ 19/9 independent of d, and after considering the T dependence of all quantities, we obtain ρD T

η ∼ 23/9 and ρD T for the Fermi surface coupled to U(1) gauge field with dynamical critical exponent

z = 3.

For this Fermi surface coupled to U(1) gauge field, we also have [161, 248]

αQ(0) ∼ T −2/3, κ¯Q(0) ∼ T 1/3. (C.36)

Thus, the correlators of thermal and thermoelectric noise in (3.8) are smaller than the correlators

272 of charge and viscous noise at low temperatures, and may be ignored. As far as corrections from

thermal current flow to our results are concerned, we observe that from (C.30) the diffusion constant

Q(0) Dσ associated with the microscopic conductivity σ always multiplies additional powers of k in

the integrands, and thus only provides subleading contributions at large values of d. This is also true

for contributions of diffusion constants associated with the microscopic thermal and thermoelectric

conductivities κ¯Q(0) and αQ(0). Thus, at large separations, thermal currents influence the charge

current flow in the dragged layer only through the entropy density, given by S(0) ∼ T 2/3 [31, 161], which, by virtue of vanishing at low temperatures, can only provide corrections to our results that

scale as subleading powers of T . All this may be verified by a very lengthy brute-force computation, which we choose not to include as it is not important for our main results.

For a general dynamical critical exponent z, the arguments of [161, 162] imply that

( ) ( ) E 2/z e2 E 2/z Q(0) ∝ F (0) ∼ (0) ∝ F ℏ 2 σ , η ζ kF . (C.37) kBT ℏ kBT

∼ σ ∼ 1−2/z 4 ≫ ∼ σ ∼ 1+10/(3z) ≪ We then get that ρD ρD T ln T for d dc, and ρD ρD T for d dc. Thus

the curvature of the ρD vs T plot (Fig. 3.1) still changes from positive to negative as d is increased,

and this change is more pronounced for z closer to 2 than to 3. In the HLR theory, z ranges between

2 (For long-range Coulomb interactions of composite fermions) and 3 (For short-range interactions of composite fermions). The scaling of ρD with T in the hydrodynamic regime at large d can thus

possibly yield some insight into the type of composite-fermion interactions leading to non-Fermi

liquid behavior.

273 C.4 Remarks on Coulomb drag in the ν = 1/2 quantum Hall state

This section serves to provide a brief summery of the main results from [179]. Consider a double-

layer system of quantum Hall states at half-filling. We want to study Coulomb drag in this setting by

adopting the theoretical framework of composite fermions (CF) based on the seminal work of HLR.

We assume that both layers are very clean so that the intralayer mean free path lcf of the composite

fermions is limited only by the scattering off the gauge field. For the case when lcf is large, drag was studied by Ussishkin and Stern [179]. Large lcf implies “collisionless” transport with respect to intralayer collisions. In this case drag resistivity can be described by the conventional expression

∫ ρ ∑ ∞ dω ρ = Q q2|W |2(ImΠ )2. (C.38) D 8π2T n2 2(ω/2T ) q,ω q,ω q 0 sinh

The screened interlayer potential can be written in the form

(V + U)/2 (V − U)/2 W = − , (C.39) 1 + Π(V + U) 1 + Π(V − U)

2 −qd where V = 2πe /εq and U = V e . The polarization operator Πq,ω entering the drag formula should be computed from the CF picture by the following prescription. According to HLR theory one starts out from the matrix equation

(Πˆ e)−1 = Cˆ + (Πˆ cf )−1, (C.40)

where Πˆ e is the electronic single layer response function, whereas Πˆ cf is the polarization (density-

density and current-current) of composite fermions, and Cˆ is the matrix of attached flux. Explicitly

274 we have      0 2πiϕ/q   Πcf 0    cf  00  Cˆ =   , Πˆ =   . (C.41) − cf 2πiϕ/q 0 0 Π11

In the limit of q/kF ≪ 1 and also ω ≪ vF q, the random-phase-approximation results for density and current responses (including Chern-Simons contributions) of composite fermions are

m q2 iωk Πcf ≈ , Πcf ≈ − + F . (C.42) 00 2π 11 24πm 2πq

cf cf The finite value of Π00 reflects the compressibility of the system, whereas Π11 reflects Landau

diamagnetism (the real part) and Landau damping (the imaginary part). Inverting Πe and taking its

density component gives

cf 3 e Π00 χq Πq,ω = Π = = . (C.43) 00 − cf cf 2 3 − 2 1 Π00Π11(2πϕ/q) q 2πiϕ χωkF

It will be convenient for us to introduce the dimensionless variables x = qd and y = ω/T such that

3 χx 2 3 Πx,y = 3 , αT = 2πϕ χT kF d . (C.44) x − iαT y

If we introduce an inverse Thomas-Fermi screening radius of composite fermions as κ = 2πe2χ/ε,

2 2 then αT can be rewritten as αT = ϕ (κd)(kF d)T/(e /εd). To proceed with the computation of ρD we notice that

−1 U Im(Π)|W | = −Im(Π ) . (C.45) (Π−1 + V + U)(Π−1 + V − U)

275 In terms of dimensionless variables

κd e−x α y U = , ImΠ−1 = − T , (C.46) χ x χx3 U χκdx5e−x −1 −1 = 3 2 −x 3 2 −x . (Π + V + U)(Π + V − U) [x − iαT y + κdx (1 + e )][x − iαT y + κdx (1 − e )] (C.47)

Now setting up an integral for the drag resistivity in dimensionless notation we obtain the following

expression

∫ 2 2 ∞ 2 ρD αT (κd) y dy = 4 2 2 2 ρQ 32π (nd ) sinh (y/2) ∫ 0 ∞ x7e−2xdx × . (C.48) | 3 − 2 −x |2| 3 − 2 − −x |2 0 x iαT y + κdx (1 + e ) x iαT y + κdx (1 e )

At the lowest temperatures T → 0 we have αT ≪ 1. Coulomb screening and thermal factors set the

typical scales of momentum and energy x ∼ y ∼ 1, however the pole structure of the denominator √ 3 is dominated by the momentum range x ∼ αT ≪ 1. Because of that we can make the following

3 2 x 2 3 2 x 3 approximations: x −iαT y+κdx (1+e ) ≈ 2κdx , x −iαT y+κdx (1−e ) ≈ (1+κd)x −iαT y

and e−2x ≈ 1. As a result, the previous expression simplifies to

∫ ∫ ρ α2 1 ∞ y2dy ∞ x3dx D = T . (C.49) 4 2 2 2 | 3 − |2 ρQ 128π (nd ) 0 sinh (y/2) 0 (1 + κd)x iαT y

Upon rescaling of x these integrals can be brought to the form

( ) ∫ ∫ 4/3 ∞ 4/3 ∞ 3 ρD 1 1 αT y dy x dx = 4 2 2 2 6 , (C.50) ρQ 128π (nd ) 1 + κd 0 sinh (y/2) 0 x + 1

276 √ where the x-integral is equal to π/(3 3) while the y-integral is equal to 4Γ(7/3)ζ(4/3). Combining √ all factors and using kF = 4πn we find

( ) ( ) 4/3 2 ρD Γ(7/3)√ζ(4/3) T e 2 1 = ,T0 = 2 (kF d) 1 + . (C.51) ρQ 6π 3 T0 ϕ εd κd

The length scale for the Coulomb potential induced on one layer by density fluctuations in the

other layer is roughly given by the interlayer separation d. For our hydrodynamic analysis to be

applicable, d must be much larger than the intralayer mean free path lcf . Thus, at small values of

4/3 4 d,(C.51) is more likely to be applicable. Since ρD falls off with d only as d in (C.51), but as d in

the hydrodynamic result (C.35), the measured spacing dependence of ρD can also possibly be used

4/3>1 to deduce the pertinent transport regime. Note that the T dependence of ρD is T in (C.51), so

the curvature of the ρD vs T plot should still switch from positive to negative as d is increased even

if there is a crossover from the collisionless to the hydrodynamic regime.

277 D Appendices to Chapter 4

D.1 Optical viscosity: two-loop computations

The two-loop self-energy correction to the stress tensor autocorrelation function is given by

∫ ( ) dd+1k ⟨T T ⟩ (iω) = 2N k2 tr γ G (k + q)γ G (k)Σ (k)G (k) xy xy SE (2π)d+1 y x 0 x 0 1 0 ∫ ( ) dd+1k µ 2ϵ/3 2δ2K2 + K · (K + Q)(δ2 − K2) = 4e4/3u ϵ−1 k2 k( k , Σ,0 d+1 y | | 2 2 2 2 2 (2π) K (K + Q) + δk)(K + δk) (D.1)

278 where we only kept the pole contribution to the self-energy and set ϵ = 0 in the prefactor uΣ,ϵ=0 =

(2 · 61/3π)−1. The self-energy correction can be computed using Feynman parameters. The integral

is first rewritten as

∫ ∫ d+1 1 − 2 2 · 2 − 2 2 ϵ 2 −1 d k 2 1 x 2δkK + K (K + Q)(δk K ) ⟨ ⟩ 3 [ ] TxyTxy SE(iω) = 4(e µ ) uΣ,0ϵ d+1 ky dx 2ϵ 3 . (2π) | | 3 2 − 2 2 0 K x(K + Q) + (1 x)K + δk (D.2)

Eliminating ky from the fraction by a variable shift of kx and subsequent integration over kx yield

∫ ∫ ∫ d−1 1 − [ 2 · Γ(3) 2 ϵ 2 −1 dky 2 d K 1 x 3K + K Q 3 × = (e µ ) uΣ,0ϵ ky d−1 dx 2ϵ [ ] 3 4 2π (2π) 0 |K| 3 K2 + x(2K · Q + Q2) 2 2 2 · ] − 3K (K + K Q) [ ] 5 . (D.3) K2 + x(2K · Q + Q2) 2

Again using Feynman parameters to rewrite the products in the integrand, we obtain

∫ ∫ ∫ ∫ d−1 1 1 Γ(3) 2 − dky d K 2 ϵ 3 1 2 = ϵ (e µ ) uΣ,0ϵ ky d−1 dx dy 4Γ( 3 ) 2π (2π) 0 0 [ 9+2ϵ ϵ −1 1 2 Γ( ) (1 − x)y 3 (1 − y) 2 (3K + K · Q) × 6 [ 3 3 ϵ 2 − · 2 2 + 3 Γ( 2 ) K + x(1 y)(2K Q + Q )] 15+2ϵ ϵ −1 3 2 2 ] Γ( ) 3(1 − x)y 3 (1 − y) 2 K (K + K · Q) − 6 [ ] . (D.4) 5 5 + ϵ Γ( 2 ) K2 + x(1 − y)(2K · Q + Q2) 2 3

Completing squares in the denominator as

K2 + x(1 − y)(2K · Q + Q2) = (K + x(1 − y)Q)2 + x(1 − y)(1 − x + xy)Q2, (D.5)

279 shifting K → K−x(1−y)Q, and neglecting terms that vanish due to symmetries when performing the K-integration, we obtain

∫ ∫ ∫ ∫ d−1 1 1 Γ(3) 2 ϵ 2 −1 dky 2 d K ϵ −1 3 − 3 = ϵ (e µ ) uΣ,0ϵ ky d−1 dx dy(1 x)y 4Γ( 3 ) 2π (2π) 0 0 { 9+2ϵ 2 − − − − 2 Γ( 6 ) 1 3K x(1 y)(1 3x(1 y))Q × (1 − y) 2 [ ] 3 3 + ϵ Γ( 2 ) K2 + x(1 − y)(1 − y + xy)Q2 2 3 15+2ϵ 3 [ Γ( ) 3(1 − y) 2 − 6 [ ] K4 − x(1 − y)(1 − 2x(1 − y))K2Q2 5 5 + ϵ Γ( 2 ) K2 + x(1 − y)(1 − x + xy)Q2 2 3 ]} − 2x(1 − y)(1 − 2x(1 − y))(K · Q)2 − x3(1 − y)3(1 − x(1 − y))Q4

(D.6)

The remaining integrals can easily be computed using Mathematica. First integrating over K and

subsequently over x and y, the pole contribution to the two-loop self-energy correction reads

∫ ( ) 2ϵ/3 4/3 −1 dky 2 1 −ϵ µ ⟨T T ⟩ (iω) = e ϵ k |ω| 2 a , (D.7) xy xy SE 2π y |ω| Σ,0

where aΣ,0 = u1Loop,0uΣ,0, after setting ϵ to zero in the numerical prefactors.

D.2 Relating conductivities and viscosities using Ward identities

The result in the main text, that the optical viscosity and optical conductivity scale in the same way, is not consistent with hyperscaling with an effectively reduced dimension. In order to substantiate this result, in the following we establish relations between the two transport quantities based on

Ward identities.

The action for the patch theory of the Ising-nematic QCP in d = 2, Eq. (4.11), is invariant under

280 an emergent rotational symmetry [28],

′ Φ(q0, qx, qy) → Φ (q0, qx, qy) = Φ(q0, qx − Θqy, qy) (D.8) ( ) Θ2 Θ ψ˜ (k , k , k ) → ψ˜′ (k , k , k ) = ψ˜ k , k − Θk − s , k + s . (D.9) sj 0 x y sj 0 x y sj 0 x y 4 y 2

We will show that this symmetry restricts the scaling behavior of transport properties as a function of frequency. Starting from this transformation law, we derive a Ward identity for the generating functional of connected correlation functions [349],

G[η†, η, ϕ] = ln Z[η†, η, ϕ] (D.10) ∫ ∫ ∑ † † ∫ † † −S[ψ˜†,ψ,˜ Φ]− (ψ˜ (k)η (k)+η (k)ψ˜ (k))− Φ(−q)ϕ(q) Z[η , η, ϕ] = D(ψ˜ , ψ˜)D(Φ)e k s,j sj sj sj sj q , (D.11)

where η(†) and ϕ are Grassmann and real source fields, respectively. Invariance under the above

rotational symmetry implies

G[η′†, η′, ϕ′] = G[η†, η, ϕ], (D.12) where the source fields transform as the physical fields. Differentiation with respect to Θ yields

d G[η′†, η′, ϕ′] = 0, (D.13) dΘ

281 which leads to the functional Ward identity

∫ 3 ∑{[( ) † d k † s † δZ[η , η, ϕ] k ∂ η (k) − ∂ η (k) (2π)3 y kx sj 2 ky sj † s,j δηsj(k) ∫ ( ) † ] † } − s δZ[η , η, ϕ] − δZ[η , η, ϕ] + ky∂kx ηsj(k) ∂ky ηsj(k) qy∂qx ϕ(q) = 0. (D.14) 2 δηsj(k) q δϕ(q)

In the following we are only interested in Ward identities for fermionic correlation functions and

thus set ϕ = 0 from the outset.

As an example how this functional Ward identity restricts correlation functions, we derive the

Ward identity that follows from rotational symmetry for the fermionic Green’s function. After

computing suitable functional derivatives, we obtain

( s ) δ2Z[η†, η, 0] ( s ) − − − py∂px ∂py † = py∂px ∂py Gs(p) = 0, (D.15) η=η†=0 2 ∂ηsj(p)∂ηsj(p) 2

where Gs(p) is the full fermionic Green’s function. This is a partial differential equation for the

momentum dependence of the latter. It can easily be verified that the Ward identity is fulfilled for

2 Gs(p) = Gs(p0, spx + py), as expected.

For q = q0e0 ≠ 0, the current-current correlation functions for the chiral current can be written

282 as

∫ ∫ 3 3 ′ ∑ d k d k † † ′ ′ ⟨J (q)J (−q)⟩ = ⟨ψ˜ (k + q)ψ˜ (k)ψ˜ ′ ′ (k − q)ψ˜ ′ ′ (k )⟩ x x (2π)3 (2π)3 js js j s j s j,s,j′,s′ ∫ ∫ (D.16) d3k d3k′ ∑ δ4Z[η†, η, 0] = Z−1 3 3 † ′ † ′ † (2π) (2π) ′ ′ − η=η =0 j,s,j′,s′ δηjs(k + q)δηjs(k)δηj s (k q)δηj′s′ (k )

⟨Jy(q)Jy(−q)⟩ = 4⟨Txy(q)Txy(−q)⟩ ∫ ∫ 3 3 ′ ∑ d k d k ′ ′ † † ′ ′ = 4 ss k k ⟨ψ˜ (k + q)ψ˜ (k)ψ˜ ′ ′ (k − q)ψ˜ ′ ′ (k )⟩ (2π)3 (2π)3 y y js js j s j s j,s,j′,s′ ∫ ∫ d3k d3k′ ∑ δ4Z[η†, η, 0] = 4Z−1 ss′k k′ . 3 3 y y † ′ † ′ † (2π) (2π) ′ ′ − η=η =0 j,s,j′,s′ δηjs(k + q)δηjs(k)δηj s (k q)δηj′s′ (k )

(D.17)

Applying functional derivatives to the functional Ward identity Eq. (D.14), we obtain a Ward iden-

tity for two-particle Green’s functions,

[ ′ ] 4 † s ′ s δ Z[η , η, 0] (k ∂ − ∂ )+(k ∂ ′ − ∂ ′ ) = 0. (D.18) y kx ky y kx ky † ′ † ′ † 2 2 ′ ′ − η=η =0 δηjs(k + q)δηjs(k)δηj s (k q)δηj′s′ (k )

Using the method of characteristics, we can show that this Ward identity restricts the dependence

of two-particle Green’s function on spatial momenta as

4 † δ Z[η , η, 0] ′ 2 ′ ′ ′2 ′ ′ = F ′ ′ (k , k , q ; sk +k , s k +k , s k −sk ), † ′ † ′ † js;j s 0 0 0 x y x y y y ′ ′ − η=η =0 δηjs(k + q)δηjs(k)δηj s (k q)δηj′s′ (k ) (D.19)

analogously to the Ward identity for the one-particle Green’s function.

Inserting this result in Eqs. (D.16) and (D.17), shifting and renaming integration variables, we

283 obtain for the Jx correlator

∫ ∫ 3 3 ′ ∑ −1 d k d k ′ 2 ′ ′ ′2 ′ ′ ⟨J (q )J (−q )⟩ = Z F ′ ′ (k , k , q ; sk + k , s k + k , s k − sk ) x 0 x 0 (2π)3 (2π)3 js;j s 0 0 0 x y x y y y j,s,j′,s′ ∫ ∫ 3 3 ′ ∑ −1 d k d k ′ ′ = Z F ′ ′ (k , k , q ; k , k , k ). (D.20) (2π)3 (2π)3 js;j s 0 0 0 x x y j,s,j′,s′

′ Note that ky does not appear in the integrand. For the Jy correlator we obtain

⟨Jy(q0)Jy(−q0)⟩ ∫ ∫ 3 3 ′ ∑ −1 d k d k ′ ′ ′ 2 ′ ′ ′2 ′ ′ = 4Z ss k k F ′ ′ (k , k , q ; sk + k , s k + k , s k − sk ) (2π)3 (2π)3 y y js;j s 0 0 0 x y x y y y j,s,j′,s′ ∫ ∫ 3 3 ′ ∑ −1 d k d k ′ ′ ′ ′ = 4Z k k F ′ ′ (k , k , q ; k , k , k − k ) (2π)3 (2π)3 y y js;j s 0 0 0 x x y y j,s,j′,s′ ∫ ∫ 3 3 ′ ∑ −1 d k d k ′ ′ ′ ′ = 2Z (k + k )k F ′ ′ (k , k , q ; k , k , k ) (2π)3 (2π)3 y y y js;j s 0 0 0 x x y j,s,j′,s′ ∫ ∫ 3 3 ′ ∑ −1 d k d k ′ ′ ′ ′ + 2Z k (k + k )F ′ ′ (k , k , q ; k , k , −k ) (D.21) (2π)3 (2π)3 y y y js;j s 0 0 0 x x y j,s,j′,s′

→ ′ ′ → ′ where in the last step we shifted ky ky + ky and ky ky + ky in the first and second term,

′ → − ′ ↔ ′ respectively. Replacing ky ky and subsequently renaming ky ky in the second term, the

∼ ′ contributions kyky cancel and we hence obtain

∫ ∫ 3 3 ′ ∑ −1 d k d k ′2 ′ ′ = 4Z k F ′ ′ (k , k , q ; k , k , k ). (D.22) (2π)3 (2π)3 y js;j s 0 0 0 x x y j,s,j′,s′

Using the Ward identity for the emergent rotational symmetry of the patch theory, we have thus

284 established that

∫ ∫ dky dky (2π) (2π) ⟨Jx(q0)Jx(−q0)⟩ = ∫ ⟨Jy(q0)Jy(−q0)⟩ = ∫ ⟨Txy(q0)Txy(−q0)⟩. (D.23) dky 2 dky 2 4 (2π) ky (2π) ky

As the Ward identity imposes restrictions only on the momentum dependence of the two-particle

Green’s function, this result is also valid in d = 5/2 − ϵ. The above result implies that the optical viscosity and the optical (chiral) conductivity have the same frequency dependence,

σ(ω) ∼ η(ω). (D.24)

Using the results from Ref. [161], we obtain

ϵ=1/2 σ(ω) ∼ η(ω) ∼ ω−1/2−ϵ/3 = ω−2/3, (D.25)

in agreement with the field theoretic result in Eq. (4.33).

D.3 Contributions from the full Fermi surface

The result Eq. (D.25) for the scaling of the optical viscosity was derived in the patch theory. The

question arises whether the scaling could be different for the full Fermi surface, for example due to

some preferred direction. In this section we show that this is not the case, using d = 2 for simplicity.

We first analyze how the Txy correlator transforms under the emergent rotation symmetry of the

285 patch theory [28]. In d = 2, Txy is given by

∑ ∫ ( ) qy † T (q) = k + ψ˜ (k + q)σ ′ ψ˜ ′ (k). (D.26) xy y 2 sj z,ss s j j,s,s′ k

We obtain

∫ ( ) ⟨ ⟩ q=ωe0 − 2 TxyTxy 1Loop(q) = N ky tr G0(k0 + ω, k)G0(k0, k) (D.27) k

for the correlation function, where we exploited in the last step that G0 and σz commute. Rotation of the Fermi momentum, with respect to which the patch theory is defined, by a small angle θ yields

∫ ( ) ∑ θ 2 ⟨T T ⟩ (q) = −N k + s G (k + ω, k)G (k , k), (D.28) xy xy 1Loop y 2 0,s 0 0,s 0 k s where

θ2 θ k → k − θk − s , k → k + s . (D.29) x x y 4 y y 2

The Green’s functions are independent of θ due to the emergent rotation symmetry. We can there-

→ − 2 fore eliminate ky from the Green’s functions by shifting kx kx sky. Then θ only appears in

the integrand of the ky integral, which is just a multiplicative prefactor, and can be eliminated by

→ − θ shifting ky ky s 2 . Nearby patches thus contribute equally to the Txy correlator.

This result can be complemented by an analysis of the stress tensor correlation function for a

286 continuum model. For an isotropic system we can start from the Lagrangian

† † † L(x) = ψ (x)∂τ ψ(x) + ∇ψ (x) · ∇ψ(x) − µψ (x)ψ(x), (D.30)

where we omitted the interaction and the bosonic contribution. The xy-component of the stress tensor reads ∫ ( ) † Txy(q) = (kx + qx)ky + (ky + qy)kx ψ (k + q)ψ(k). (D.31) k

At one-loop level, the Txy autocorrelation function for q = 0 is then given by

∫ ⟨ ⟩ − 2 2 TxyTxy 1Loop(iω) = 4 kxkyG0(k + q)G0(k) (D.32) k

where q = ωe0.

We can subdivide the vicinity of the Fermi surface into (finite) patches, which are labeled by ϕ,

and obtain

∑ ∫ ⟨ ⟩ − ′ − ′ 2 ′ ′ 2 TxyTxy 1Loop(iω) = 4 (kx cos ϕ ky sin ϕ) (kx sin ϕ + ky cos ϕ) ′ Patches k ( ) × ′ ′ ′ ′ tr G0(k0 + ω, k )G0(k0, k ) , (D.33)

where the integral over k′ is over a specific patch. The sum over patches (or ϕ-integration) sums up the contributions from individual patches. The Green’s functions do not depend on ϕ because they are the same in each local patch coordinate system (Fig. D.1) and are just given by the patch

′ → ′ theory action in the supplement. Shifting kx kF + kx in order to make the Fermi momentum

287 explicit, we obtain

( ) 1 2 (k′ cos ϕ − k′ sin ϕ)2(k′ sin ϕ + k′ cos ϕ)2 → 2k k′ cos(2ϕ) + k2 sin(2ϕ) − k′ 2 sin(2ϕ) x y x y 4 F y F y ( ) 1 + k′ 2 6k k′ sin(4ϕ) + 6k2 sin2(2ϕ) + k′ 2(3 cos(4ϕ) + 1) 4 x F y F y ( ) 1 + k′ 3k2 k′ sin(4ϕ) + k k′ 2(3 cos(4ϕ) + 1) + 2k3 sin2(2ϕ) − k′ 3 sin(4ϕ) 2 x F y F y F y ( ) ′ 3 ′ ′ 4 2 2 + kx sin(2ϕ) kF sin(2ϕ) + ky cos(2ϕ) + kx sin ϕ cos ϕ. (D.34)

′ The terms in the first line of the right hand side do not depend on kx and yield the scaling that we

′ determined from the patch theory as ky and kF do not scale. The terms on the other lines contain

′ ∼ 1/z additional powers of kx ω and are hence subleading. The above argument takes care of the

Figure D.1: Transformation of coordinates used to determine the contribution of different patches to the Txy - Txy correlator.

′ ′ 1-loop and self-energy corrections. In the vertex corrections, the additional powers of kx and ky do

not influence the absence of poles in ϵ−1. Hence all patches contribute the same scaling at leading

order in ω.

The expressions in Eqs. (D.27) and (D.32) are directly related only for the patches in the kx or

2 2 the ky direction. In the former case, evaluating the factor kxky in the integrand close to the Fermi

′ 2 ′ 2 ≈ 2 ′ 2 surface yields (kF + kx) ky kF ky . After rescaling of momentum variables, this yields the

288 2 factor of ky that appears in the stress tensor correlation function of the patch theory in Eq. (D.27).

For other directions additional terms appear, which are not present in the patch theory, for example

4 terms in Eq. (D.34) which are proportional to kF . As kF and ky do not scale, such terms are equally relevant to the terms that appear in the patch theory and thus do not change the scaling behavior. The argument employing the emergent rotational symmetry does not generate such terms, but nevertheless leads to the correct scaling behavior.

289 E Appendices to Chapter 5

E.1 Effects of ‘Pair-hopping’ and bilinear terms on the marginal-

Fermi liquid

We consider the effects of the ‘pair-hopping’ term (5.18) on the MFL as T → 0. With the Hamil-

tonian given by (5.18), the Dyson equations are given by

M M Σ(τ) = −J 2G2(τ)G(−τ) − g2G(τ)Gc(τ)Gc(−τ) − η2G(−τ)(Gc(τ))2, N N 1 G(iωn) = , iωn + µ − Σ(iωn)

290 Σc(τ) = −g2Gc(τ)G(τ)G(−τ) − η2Gc(−τ)(G(τ))2, ∫ d c d k 1 G (iωn) = d c . (E.1) (2π) iωn − ϵk + µ − Σ (iωn)

If µ = 0, the exact relations G(τ) = −G(−τ) and Gc(τ) = −Gc(−τ) imply that the only effect of the pair-hopping term on the physics considered in the main text in all regimes is just a redefinition of g, with g → (g2 + η2)1/2.

As long as the bandwidth is large, i.e. t ≫ g, η, J,(5.9) is still valid. Following the same

procedure as we did in Sec. 5.3.1, and using G(τ) given by (5.11), we obtain

( ) − 2 | | γE 1 c ig ν(0) ωn e Σ (iωn → 0) = ωn ln 2J cosh1/2(2πE)π3/2 J ( ( ) ) − η2ν(0) cosh1/2(2πE) ω |ω |eγE 1 + i n ln n − tanh(2πE) + O(ω ). (E.2) 2π3/2 J J n

This is clearly a marginal-Fermi liquid with an additional chemical potential correction

η2ν(0) cosh1/2(2πE) δµ = tanh(2πE) ≪ Λ, (E.3) 2π3/2 which leads to a harmless small change in the size of the conduction electron Fermi surface, as the

numbers of c and f electrons are no longer independently conserved (but their sum is conserved).

There is also a back-reaction to the SYK islands

M M Σ(˜ τ) = − g2G(τ)Gc(τ)Gc(−τ) − η2G(−τ)(Gc(τ))2, (E.4) N N 2 2 E ˜ Mg (ν(0)) J sinh(π ) Σ(iωn → 0) = √ + O(ωn), (E.5) 3 2Nπ9/4 cosh1/4(2πE)

291 which is again a chemical potential correction plus irrelevant frequency-dependent corrections.

This chemical potential correction actually changes E, which is no longer a conserved quantity,

and is determined by the condition Re[Σ(iωn → 0)] = µ + δµ.

We also briefly discuss qualitatively the effects of certain fermion bilinears in (5.3). Terms bi-

linear in the f’s destroy their SYK behavior and non-zero entropy as T → 0. The c’s then scatter

off essentially non-interacting random-matrix islands, with G(iωn) ∼ isgn(ωn). This leads to

c ∼ 2 → Im[ΣR(0)] T , and the c’s hence realize a weakly-interacting disordered Fermi liquid as T 0.

However, if the coefficients of the f-bilinears are small, then their SYK behavior is restored for temperatures larger than a small energy scale Ec [42]. Hence, the marginal-Fermi liquid behavior

of the c’s is also restored for T > Ec.

The effects of bilinears which hybridize c’s and f’s (such as c†f) were disussed in Ref. [235]. In √ → ∞ c ∼ the N limit, these lead to Im[ΣR(0)] 1/ T when the f’s are described by SYK models.

This is more relevant than the MFL self-energy (∼ T ) at low T , but less relevant at high T . Thus,

once again, if the coefficients of these bilinears are small, then the MFL self-energy will dominate

above a certain temperature scale, and the MFL behavior will be restored.

E.2 Boltzmann equation for the marginal-Fermi liquid

We provide a derivation of (5.33). We follow the notation, style, and mechanics of Chapter 5 of

Ref. [105]. The general off-shell Boltzmann equation for modes close to the isotropic Fermi surface

(|p| ≈ pF ; we do not use boldface for momentum-space vectors) is given by

− |∇ | ◦ c − c ◦ − ◦ c [(i∂t + vF + AE + AB ) ,F ] = ΣK (ΣR F F ΣA), (E.6)

292 where F (t, r, p, ω) = 1−2(nf (ω)+δn(t, r, p, ω)) is a parameterization of the distribution function,

AE(t) and AB(r) are parts of the electromagnetic vector potential giving rise to the uniform electric and magnetic fields respectively, with −dAE(t)/dt = E(t) and ∇ × AB(r) = Bzˆ (∇ denotes the

c spatial gradient). ΣR,A,K are the retarded, advanced, and Keldysh components of the conduction

electron self-energy respectively. The equation (E.6) follows from the Dyson equation for two- point functions on the Keldysh contour [105], and hence is exact due to the large M,N limits. The

◦ denotes the convolution

∫ 2 Z = X ◦ Y ⇒ Z(t1, r1, t2, r2) = dt3d r3 X(t1, r1, t3, r3)Y (t3, r3, t2, r2), (E.7)

in the two-coordinate representation, and the [.,.] denotes a commutator. We will however mostly

use the central-relative coordinate representation instead, with p, ω being Fourier transforms of the

relative coordinate r1 − r2, t1 − t2, and r, t denoting the central coordinate (r1 + r2)/2, (t1 + t2)/2; this convolution can then be appropriately re-expressed in this representation following Ref. [105].

We then use a coordinate remapping k = p + AB(r) [350, 351] to redefine F (t, r, p, ω) =

1 − 2(nf (ω) + δn(t, r, p, ω)) ⇒ F (t, k, ω) = 1 − 2(nf (ω) + δn(t, k, ω)). This is valid as long as

the Fermi energy is large enough to make effects of Landau quantization insignificant at the fields in

question. The only r dependence in F then is fictitious, coming from the r dependence of AB, and

should not affect physical results for spatially uniform transport quantities due to gauge-invariance.

It is now absorbed into an implicit r dependence in k.

We consider the part of (E.6) proportional to the infinitesimal E(t). Because of the isotropy of

the Fermi surface and the scattering, we then use the ansatz δn(t, k, ω) = k · φ(t, ω). We use a

first-order gradient expansion in spatial and time derivatives with respect to the central coordinate,

293 which is justified by the spatial uniformity of E(t) and B, and the slow temporal variation of E(t).

The change in the momentum-integrated Keldysh conduction electron Green’s function caused by

E(t) through δn then is [105]

R,A ∇ ∇ · as Gc are isotropic. We have used δn(t, k, ω) = AB(r) ∂kδn(t, k, ω), due to the implicit r

dependence in k. The retarded and advanced conduction electron Green’s functions are not changed

by the applied electric field, as they are only influenced by the change in the distribution δn through

the self-energies [105], which as we show below, are unaffected by the applied electric field.

On the Keldysh contour, the conduction electron self-energy is given by, analogous to (5.6),

c 2 c Σ (t1, t2) = −g G (t1, t2)G(t1, t2)G(t2, t1),

c − 2 c Σ>,<(t1, t2) = g G>,<(t1, t2)G>,<(t1, t2)G<,>(t2, t1). (E.9)

Using the standard relations between the >, < representation and the R, A, K representation [105,

352], the changes in the conduction electron self-energies due to δn are then given by

g2 δΣc (t , t ) = − θ(t − t )δGc (t , t )(G (t , t )G (t , t ) + G (t , t )G (t , t )), R 1 2 4 1 2 K 1 2 K 1 2 A 2 1 K 2 1 R 1 2 g2 δΣc (t , t ) = − θ(t − t )δGc (t , t )(G (t , t )G (t , t ) + G (t , t )G (t , t )), A 1 2 4 2 1 K 1 2 K 1 2 R 2 1 K 2 1 A 1 2

294 g2 δΣc (t , t ) = − δGc (t , t )(G (t , t )G (t , t ) + G (t , t )G (t , t )), (t > t ), K 1 2 4 K 1 2 K 1 2 K 2 1 R 1 2 A 2 1 1 2 g2 δΣc (t , t ) = − δGc (t , t )(G (t , t )G (t , t ) + G (t , t )G (t , t )), (t < t ), (E.10) K 1 2 4 K 1 2 K 1 2 K 2 1 A 1 2 R 2 1 1 2

which vanish due to (E.8). Here, GR,A,K denote the island electron Green’s functions at equilibrium.

Similarly, for the islands, we also get δΣR,A,K = 0, for the same reason.

O c ◦ − ◦ c The (E) part of the RHS of (E.6) then is 2(ΣR δn δn ΣA). Using the p, k, r-independence

c of the by definition t-independent equilibrium self-energies ΣR,A,K , and a first-order gradient ex-

pansion in central time derivatives, the RHS of (E.6) reduces to [105]

c c 4iIm[ΣR(ω)]δn(t, k, ω) + 2i∂ωRe[ΣR(ω)]∂tδn(t, k, ω). (E.11)

We now turn to the part of the LHS of (E.6) proportional to E(t). Following Sec. 5.7 of Ref. [105],

and noting that the Wigner transform of ∇ + AB(r) is k, it reduces in the first-order gradient

expansion in central spatial and time derivatives to

2i∂tδn(t, k, ω) ( ) − | | ′ ∇| | · − | | · ∇ · + 2i vF ∂t k + AE(t) nf (ω) + vF k ∂kδn(t, k, ω) vF ∂k k AB(r) ∂kδn(t, k, ω) ,

(E.12)

After some algebra, this further reduces to

· ˆ ′ B ˆ × · 2i∂tδn(t, k, ω) + 2ivF E(t) knf (ω) + 2ivF (k zˆ) ∂kδn(t, k, ω). (E.13)

Then, combining this with (E.11), we recover (5.33). The solution to (5.33) then shows our ansatz

295 δn(t, k, ω) = k · φ(t, ω) to be self-consistent.

296 F Appendices to Chapter 6

F.1 Derivation of gap equations

We derive the combined Dyson and gap equations from the large-N saddle point action. For Model

1, the disorder averaged action can be written as [37, 41, 42, 286]

∫ [ ] β ∑ ( ) ∑ ( ) † − ↔ − † ↔ S = dτ aim(∂τ µ)aim + (a b) t aimain + (a b) + H.c. 0 m,i ⟨m,n⟩,i ∫ K2 β ∑ ( ) − N dτdτ ′ G2 (τ, τ ′)G2 (τ ′, τ) + (a ↔ b) 4 am am 0 m

297 ∫ [ ( ) ] β ∑ ∑ 1 † − N dτdτ ′ Σ (τ, τ ′) G (τ ′, τ) + a (τ)a (τ ′) + (a ↔ b) am am N im im 0 m i ∫ ∫ [ ( ) ] β ∑ β ∑ 1 ∑ − NU dτ ∆∗ (τ)∆ (τ) − N Ξ (τ) ∆ (τ) − a (τ)b (τ) + H.c . 0m 0m m 0m N im im 0 m 0 m i (F.1)

′ ∗ In the large-N limit, the Lagrange multiplier fields Σαm(τ, τ ) and Ξm(τ), Ξm(τ) enforce the defi-

G ∗ nitions of and ∆0, ∆0 at each site m. After integrating out the fermions, varying the action with

G ∗ respect to αm and ∆0m, ∆0m yields

′ − 2G2 ′ G ′ Σαm(τ, τ ) = K αm(τ, τ ) αm(τ , τ),

− ∗ ∗ − Ξm(τ) = U∆0m(τ), Ξm(τ) = U∆0m(τ). (F.2)

For the saddle point, we look for a uniform solution in which Σ, G, Ξ, ∆0 are independent of m and

α ≡ a, b, and where Ξ and ∆0 are constant in time. Then, additionally varying the action with the

fermions integrated out with respect to Ξ, Ξ∗ and Σ(τ, τ ′) and applying (F.2) generates the set of

equations in (6.3).

The result (6.4) for Tsc in the limit of small bandwidth may then be derived from (6.3) by sending

∆0 → 0 and ignoring the dispersion ξk. We get

∫ −1− −1 Tsc K 1 dτ G2(τ) = . (F.3) K−1 U

Using the IM Green’s function (6.2) appropriate to this limit, we obtain (6.4).

After integrating out the fermions and taking the saddle point of (F.1), fluctuations in the pairing

iθm(τ) order parameter ∆0m(τ) = ∆0e only affect the fermion determinant term. Then, the effective

298 action for the long-wavelength condensate phase fluctuations θm(τ) can be derived following the

standard procedure in Ref. [270], yielding a coefficient of (∇θ)2 and hence a superfluid stiffness

that is extensive in N. Due to the large-N limit, this implies that phase coherence for T < Tsc is established for any nonzero values of t and ∆0.

A procedure similar to the one described in this section can be applied to derive the gap equations for Model 2.

F.2 Superconducting transition energetics

In order to study the details of the IM-SC transition, we compute the free energy density at the

large-N saddle point in Model 1

∫ [ ] F ∑ ddk ω2 + ξ2 + U 2|∆ |2 = T ln n k 0 N (2π)d |G(iω , k)|−2 + U 2|∆ |2 ω n 0 ∫ n d [( √ )( √ )] d k U 2|∆ |2+ξ2/T − U 2|∆ |2+ξ2/T − T ln 1 + e 0 k 1 + e 0 k (2π)d 3 ∑ − T Σ(iω )G(iω ) + U|∆ |2. (F.4) 2 n n 0 ωn

To analyze this as a function of the order parameter ∆0, we determine G, Σ, G as functions of ∆0

using (6.3), but ignore its last line that determines ∆0 self-consistently. A plot of F/N vs |∆0| for

T ≳ Tsc and T ≲ Tsc is shown in Fig. F.1.

6 2 The free energy functional is qualitatively similar to that of ϕ theory, with F ∼ N(c2|∆0| −

4 6 c4|∆0| + c6|∆0| ), where c2 ∝ (T − Tsc). Thus, as T is lowered below Tsc there is a first-order transition from a state with zero gap to a state with a finite gap. The free energies of the gapped and

(1) gapless states actually cross each other at a temperature Tsc which is slightly larger than Tsc, but

299 - 16.6

- 16.8

- 17.0

- 17.2

- 17.4 0.0 0.1 0.2 0.3 0.4 0.5 Figure F.1: A plot of the free energy density in the IM/SC regime in Model 1, as a function of the superconducting order parameter ∆0. The values of parameters used are K = 100, Λ = 1, U = 41.34, so that Tsc = 1.

the large-N limit strongly suppresses tunneling from one local minimum to another: the tunneling

rate goes as ∼ e−N in the WKB approximation. A similar effect is also seen for Model 2.

F.3 Real-time Dyson equations

Superconductors– The real-time Green’s functions for the gapped SC phase of Model 1 can be obtained by solving the Dyson equation (6.3) on the Keldysh contour. It is given by

∫ ddk G (ω, k) G (ω) = R , R d 2 2 ∗ − (2π) 1 + U ∆0GR(ω, k)GR( ω, k)

GK (ω) = 2iIm[GR(ω)] tanh(ω/(2T )), [ ] K2 Σ (t) = − θ(t) (G (t) + G (t))2(G∗ (t) − G∗ (t)) − (G (t) − G (t))2(G∗ (t) + G∗ (t)) , R 8 K R K R K R K R [ ] K2 Σ (t > 0) = − (G (t) + G (t))2(G∗ (t) − G∗ (t)) + (G (t) − G (t))2(G∗ (t) + G∗ (t)) , K 8 K R K R K R K R − ∗ − ΣK (t < 0) = ΣK ( t > 0), [ ] + −1 GR(ω, k) = 2 tanh(ω/(2T )) 2 tanh(ω/(2T ))(ω − ξk + i0 − Re[ΣR(ω)]) − ΣK (ω) , (F.5)

where we exploited the standard simplifications for a system in equilibrium [105].

300 3.5

2.5

1.5

0.5

0 -8 -4 0 4 8

Figure F.2: A plot of the spectral function A(ω, {k : ξk = 0}) in the SC phase at of Model 1 as T → 0, for values of parameters such that the SC arises out of the IM. The spectral function is strongly peaked at the physical gap ω = ∆. The values of the parameters used are T = 0.01, K = 1000, Λ = 1, U = 269.35, corresponding to Tsc = 1 and ∆ = 8.39.

The order parameter ∆0 is first determined from the solution of the imaginary-time equations

(6.3), and inserted into the real-time equations, which are then solved iteratively, determining GR(ω)

and GR(ω). A plot of A(ω, {k : ξk = 0}) is shown in Fig. F.2, clearly showing peaks at the physical

gap ω = ∆. The same strategy can also be applied to Model 2.

Incoherent metals– In order to compute the transport properties of Model 2 described in Fig. 6.3, we need to numerically find the real-time Green’s function GR(ω) in the metallic phase. This can

be done by solving the Dyson equation (6.9) on the Keldysh contour. It reads as

[ ] J 2 Σ (t) = − θ(t) (G (t) + G (t))4(G∗ (t) − G∗ (t))3 − (G (t) − G (t))4(G∗ (t) + G∗ (t))3 R 128 K R K R K R K R [ ] K2 − θ(t) (G (t) + G (t))2(G∗ (t) − G∗ (t)) − (G (t) − G (t))2(G∗ (t) + G∗ (t)) , 8 K R K R K R K R [ ] J 2 Σ (t > 0) = − (G (t) + G (t))4(G∗ (t) − G∗ (t))3 + (G (t) − G (t))4(G∗ (t) + G∗ (t))3 K 128 K R K R K R K R [ ] K2 − (G (t) + G (t))2(G∗ (t) − G∗ (t)) + (G (t) − G (t))2(G∗ (t) + G∗ (t)) , 8 K R K R K R K R

301 − ∗ − ΣK (t < 0) = ΣK ( t > 0), 2 tanh(ω/(2T )) GR(ω) = , 2 tanh(ω/(2T ))(ω − Re[ΣR(ω)]) − ΣK (ω)

GK (ω) = 2iIm[GR(ω)] tanh(ω/(2T )). (F.6)

These equations can be solved by iteration just like their imaginary time counterparts, determining

GR(ω) and GK (ω). The real-time retarded version of the current-current correlator (6.11) is

ˆl ˆl ≪ ⟨I I ⟩R(q = 0, t) ≫ [ ] NK2 = θ(t) (G (t) + G (t))2(G∗ (t) − G∗ (t))2 − (G (t) − G (t))2(G∗ (t) + G∗ (t))2 , 4z K R K R K R K R (F.7) with the uniform DC conductivity

d ˆl ˆl σDC = lim (≪ ⟨I I ⟩R(q = 0, ω) ≫). (F.8) ω→0 dω

302 F.4 Gap equations for Model 2

Since the leading instability is to a uniform paired state, after condensing the order parameter ∆0 = ∑ ⟨ ⟩ i aimbim /N, we obtain the following gap equations in the large-N limit:

G G(iωn) (iωn) = 2 2 2 , 1 + U |∆0| |G(iωn)|

Σ(τ − τ ′) = −J 2G4(τ − τ ′)G3(τ ′ − τ) − K2G2(τ − τ ′)G(τ ′ − τ), ∑ |G(iω )|2∆ ∆ −1 − n 0 0 G (iωn) = iωn Σ(iωn),T 2 2 2 = . (F.9) 1 + U |∆0| |G(iωn)| U ωn

Sending ∆0 → 0 allows us to determine Tsc.

303 G Appendices to Chapter 7

G.1 Higgs transition from the U(1) ACL to a Z2 ACL

We consider a Higgs transition that breaks the U(1) gauge invariance down to Z2 in the ACL of

Sec. 7.5. This is expected to be a toy model of the optimal doping transition in the cuprates without

a symmetry-breaking order parameter, from the overdoped to the underdoped side [278–281]. We

304 modify the fermion-gauge field hamiltonian to

∑ ∑M ∑ [ ] ′′ 1 † † H = − tαβf eisAij f + (2MN)1/2µδαβf f , 1 (2MN)1/2 ijs iαs jβs ij iαs iαs ⟨ij⟩ αβ=1 s=

≪ αβ βα ≫ ≪ | αβ|2 ≫ 2 tijstjis = tijs = t . (G.1)

↔ − αβ We have now broken the + pseudospin symmetry since the hopping matrix elements tijs are uncorrelated between s = . However, this symmetry is restored upon disorder-average as the

αβ variances of the tijs are the same for s = . This will allow us to easily write down saddle-point equations in the higgsed phase, as the 4-Fermi term produced by disorder-averaging will not have

⟨ † ⟩ decompositions in the f+f− channel that would prevent its decomposition exclusively into the

H′′ Gi’s. As before, the addition of Maxwell terms and time components for the gauge fields to 1 is implied.

Now we add complex scalar Higgs fields Hi defined on each site i of the N-dimensional hyper-

2iθi cube into the mix. These fields are charge 2 under the U(1) gauge field, with Hi → Hie under the U(1) gauge transformation.

[ ( )] ∑ ∑M ∑ [ ] ′′ 2 † tH ∗ 2iA H = Mr|H | + g H f f − + h.c. − H H e ij + h.c. . (G.2) 2 i H i iα+ iα 2 i j i α=1 ⟨ij⟩

H′′ The addition of coupling to time components of the gauge fields to 2 is implied. The couplings of the Higgs fields to the fermions are non-random, but a large-M,N saddle-point can still be defined as was done in Ref. [353], which had non-random couplings to a superconducting order

H′′ H′′ parameter. To see this, we disorder-average the action of 1 + 2 and then expand the exponentials

305 to quadratic order as before (ignoring the screened time components of the gauge fields),

[ ] ∫ ∑ ∑M ∑ ( ) ′′ † † S = dτ fiαs(τ)(∂τ + µ)fiαs(τ) + gH fiα+(τ)Hi(τ)fiα−(τ) + h.c. i α=1 s= ∫ [( ) ] M ∑ ∑ 1 1 + t2 dτdτ ′ 1 − A2 (τ) − A2 (τ ′) + A (τ)A (τ ′) + is(A (τ) − A (τ ′)) 2N 2 ij 2 ij ij ij ij ij ⟨ ⟩ ij s= [ ] ∫ ∑ ∑ ∑M 1 † × G (τ, τ ′)G (τ ′, τ) − M dτdτ ′ Σ (τ, τ ′) G (τ ′, τ) − f (τ ′)f (τ) js is is is M iαs iαs i s= α=1 ∫ ∑ [ ] 2 2 + M dτ |∂τ Hi(τ)| + r|Hi(τ)| ∫ i t ∑ [ ( ) ] − H dτ H∗(τ) 1 + 2iA (τ) − 2A2 (τ) H (τ) + h.c. . (G.3) 2 i ij ij j ⟨ij⟩

We now integrate out the fermions and gauge fields

  ∑ ∂ + µδ(τ, τ ′) − Σ (τ, τ ′) g H (τ)δ(τ, τ ′)  ′′  τ i+ H i  S = −M Tr ln   i ∗ ′ ′ − ′ gH Hi (τ)δ(τ, τ ) ∂τ + µδ(τ, τ ) Σi−(τ, τ ) ∫ ∑ [ ] 2 2 + M dτ |∂τ Hi(τ)| + r|Hi(τ)| i[ ] 1 ∑ ∂2 + Tr ln − τ + Π˜ (τ, τ ′) + 2t (H∗(τ)H (τ) + h.c.)δ(τ, τ ′) 2 g2 ij H i j ⟨ij⟩ ∫ t ∑ − H dτ [H∗(τ)H (τ) + h.c.] 2 i j ⟨ij⟩ ∫ ∫ M ∑ ∑ ∑ ∑ + t2 dτdτ ′ G (τ, τ ′)G (τ ′, τ) − M dτdτ ′ Σ (τ, τ ′)G (τ ′, τ), 2N js is is is ⟨ij⟩ s= i s=

˜ ′ Πij(τ, τ ) [ ∫ ] M ∑ 1 = t2 G (τ ′, τ)G (τ, τ ′) − δ(τ, τ ′) dτ ′′ (G (τ, τ ′′)G (τ ′′, τ) + G (τ ′′, τ)G (τ, τ ′′)) . N is js 2 is js is js s=

(G.4)

306 where we threw out some terms that do not contribute to first-order variations at the saddle-point we will obtain. In addition to the saddle-point for Gis and Σis, this action also has a saddle-point for

Hi. Fluctuations of Hi about this saddle point are suppressed by the large-M limit. The combined saddle-point equations obtained by varying Gis, Σis and Hi about an i, s-uniform solution with

constant |H(τ)| = |H| are

∫ − 2 2 dΩm G(iωn + iΩm) G(iωn) Σ(iωn) = t G(iωn) + t T , 2 2 ˜ | |2 2π Ωm/g + Π(iΩm) + 4tH H

iωn + µ − Σ(iωn) G(iωn) = 2 2 2 , (iωn + µ − Σ(iωn)) − g |H| [ ∫ H N dω g2 H r − t + n H H − 2 − 2 | |2 M 2π (iωn + µ Σ(iωn)) gH H ∫ ] 2N dΩ t + m H = 0, 2 2 ˜ | |2 M 2π Ωm/g + Π(iΩm) + 4tH H ∫ M dω Π(˜ iΩ ) = 2t2 n G(iω )(G(iω + iΩ ) − G(iω )). (G.5) m N 2π n n m n

Saddle-points for which H is static in time with a spatially uniform magnitude but spatially varying phase are gauge-equivalent to the uniform solution, and yield the same fermion Green’s

function. For r between NtH /M and

∫ N dω g2 ≡ − − n H rc tH 2 , (G.6) M 2π (iωn + µ − Σ(iωn)) H=0

the equations (G.5) have a solution with a Higgs condensate |H| ̸= 0, with |H| vanishing as r →

rc. In this higgsed phase, the only remaining gauge redundancy is a Z2 gauge transformation of

f → −f. The condensate renders the low-energy fluctuations of the gauge fields non-singular, which causes the low-energy fermion Green’s function and self-energy to take on a random-matrix

307 - 1.199 - 0.180

- 1.200 - 0.181

- 1.201 - 0.182

- 1.202 - 0.183

- 1.203 - 0.184

0.000 0.005 0.010 0.015 0.020 0.00 0.01 0.02 0.03 0.04

Figure G.1: (a) Plot of r as a function of |H| in the higgsed phase, obtained from numerical solution of (G.5) in the 2 3 2 T → 0 limit. The orange line fits the numerical data with r = rc + h2|H| + h3|H| , so r − rc ∼ |H| as |H| → 0. 2 The values of parameters used are t = tH = g = gH = 1, 2M = N and µ = 0. (b) Plot of the free energies per fermionic degree of freedom of the |H| ̸= 0 solution (orange) and H = 0 solution (blue) of (G.5). The weak first-order behavior at very small |H| is due to a small finite T = 10−5 in the numerics, and disappears as T → 0. The values of other parameters used are the same as in (a).

form G(iωn), Σ(iωn) ∼ isgn(ωn) for gH |H| ≪ t. The reasoning behind this is the same as that for the solution of (7.43), and the random-matrix like solution at low energies can easily be verified

by solving (G.5) numerically using the MATLAB code gdHiggs.m [354]. Relative to the non-Fermi

liquid U(1) ACL phase, the low-energy fermion density of states ∼ Im[GR(ω)] is thus depleted,

akin to a ‘pseudogap’ phase. Furthermore, the resistivity in the higgsed phase, following from

DC ∼ 2 (7.48), becomes Fermi-liquid like, with ρxx ρ0 + ρ1T .

2 Fig. G.1 shows the onset of the Higgs condensate, with r − rc ∼ |H| as |H| → 0, indicating

a continuous transition with exponent ν = 1/2 as T → 0. Also shown is the comparison of free

energies of the |H| ̸= 0 solution and the H = 0 solution of (G.5) for values of r that allow for the

higgsed phase; this shows that the |H| ̸= 0 saddle point is indeed energetically favorable as T → 0.

308 H Appendices to Chapter 8

H.1 Self energies

The one-loop self energy graphs are shown in Fig. H.1. The derivation of the one-loop boson self

energy is standard [28]

∫ ∑ d2q 1 1 Π(k) = −Ne2T (2π)2 q + q2 − iq (k + q ) + (k + q )2 − i(q + k ) q x y 0 x x y y 0 0 ∫ 0 2 2 2 d q nf (qx + q ) − nf (kx + qx + (ky + qy) ) = −Ne2 y 2 2 − 2 − (2π) qy (ky + qy) + ik0 kx

309 ∫ 2 2 2 Ne dqy qy Ne |k0| = = − + Π∞ (H.1) | | 2 2 2 | | 2 ky (2π) qy + k0 8π ky

The formally infinite piece Π∞ is tuned away by the mass renormalization at the critical point, giving the expression for the boson propagator in the main text. We can further see, if we do the qx

and qy integrals before the frequency summation, that we get the same expression for Π(k) if the renormalized fermion propagator (8.8) is used instead of the bare one, implying that (8.7) and (8.8) are the solution of self-consistent “Eliashberg” equations [28], even at T ≠ 0.

Figure H.1: (a) The one-loop boson and (b) fermion self energies. These graphs are evaluated at a finite temperature. The dashed lines are bare boson propagators and solid lines are bare fermion propagators. The arrows indicate the directions of momentum flow used in the equations in the text.

The one loop fermion self energy is given by

∫ e2 ∑ d2q |q | 1 Σ(k) = T y N (2π)2 |q |3 + c |q | + m2 k + q + (k + q )2 − i(k + q ) q y b 0 x x y y 0 0 ∫0 2 ∑ ie dqy |qy| = T 3 2 sgn(k0 + q0) 2N 2π |qy| + cb|q0| + m q0 2 ∑ 2 ie sgn(k0 + q0) e T = √ T + isgn(k0) √ 1/3 | |1/3 2/3 3 3c N q0 3 3m N b q0=0̸ | | 2 ∑nk 2ie sgn(k0) 2/3 1 µ(T ) = √ T + isgn(k0) (k0 = 2πT (nk + 1/2), q0 = 2πT nq) 1/3 1/3 1/3 N 3 3cb (2π) N n =1 nq q ( ) c sgn(k ) µ(T ) c sgn(k ) |k | − πT sgn(k ) = i f 0 T 2/3H (|n |) + isgn(k ) = i f 0 T 2/3H 0 0 N 1/3 k 0 N N 1/3 2πT µ(T ) + isgn(k ) , (H.2) 0 N

310 which gives the expression for the fermion propagator in the main text.

H.2 Wightman functions

The Wightman function for two operators A, B of concern to us is

W −βH GAB(x, t) = Tr[e A(x, t)B(0, iβ/2)] ∑ −βEn −β(Em−En) −i(En−Em)(t−iβ/2) = ⟨En|B(0)|Em⟩⟨Em|A(x, 0)|En⟩e e e nm ∑ −βEn −β(Em−En)/2 −i(En−Em)t = ⟨En|B(0)|Em⟩⟨Em|A(x, 0)|En⟩e e e . (H.3) nm

W GAB(k, ω) ∑ ∫ d −ikx −βEn −β(Em−En)/2 = 2π ⟨En|B(0)|Em⟩⟨Em| d xA(x, 0)e |En⟩e e δ(ω − (En − Em)) nm ∑ ∫ d −ikx = 2π ⟨En|B(0)|Em⟩⟨Em| d xA(x, 0)e |En⟩ nm

β(En−Em) − e ∓ 1 × βEn − − e δ(ω (En Em)) − − − , (H.4) eβ(En Em)/2 ∓ e β(En Em)/2 where the − sign is for bosonic operators and + sign is for fermionic operators. Using the definition

of the spectral function

SAB(k, ω) ∑ ∫ d −ikx −βEn β(En−Em) = 2π ⟨En|B(0)|Em⟩⟨Em| d xA(x, 0)e |En⟩e δ(ω − (En − Em))(e ∓ 1), nm (H.5)

311 Figure H.2: (a), (b) The two simplest crossed ladder insertions in the Bethe-Salpeter equation. The first vanishes, and the second contributes to λL at O(1/N). (c) A higher-order “maximally crossed” diagram with boson rungs. Diagrams of this type also vanish for the same reason as (a). we have

W SAB(k, ω) GAB(k, ω) = βω (bosons), 2 sinh 2

W SAB(k, ω) GAB(k, ω) = βω (fermions). (H.6) 2 cosh 2

H.3 Higher order corrections

We consider the corrections to the ladder series of the main text coming from diagrams with crossed rungs. We show that certain diagrams with crossed boson rungs vanish, and that diagrams with crossed fermion rungs contribute to λL at higher orders in 1/N.

There are two simple types of crossed ladder insertions in the Bethe-Salpeter equation. The first is shown in Fig. H.2a and is given by

∫ d3k I (k, k′, ω) = e4 1 DW (k − k )DW (k − k′)GR(k )GR∗(k + k′ − k − ω). (H.7) 1 (2π)3 1 1 1 1

W R R∗ ′ The integral over k1x vanishes as the D ’s do not depend on k1x and G (k1)G (k+k −k1−ω) has

two simple poles both in the upper half-plane for the k1x integration. Thus this insertion contributes

nothing. Other “maximally crossed” diagrams of the same type (Fig. H.2c) also vanish for exactly

312 the same reason.

The insertion of Fig. H.2b does not vanish. However, unlike the third diagram on the right hand

side of Fig. 8.2, the flavor indices on the two sides of the insertion are not decoupled. Thus, there is no factor of N enhancement from an additional sum over flavors, and this insertion is smaller by a factor of 1/N (The integrals for this insertion are similar to the integrals for the “Box” insertion

considered in the main text and do not contain any IR divergences that enhance its value by factors

of N).

Finally, we must mention that, due to the uncontrolledness of the large N expansion, there will

be more complicated higher-loop insertions that, although naively down powers of N, will end up

contributing at the same order as the diagrams we considered in the main text. We do not know

how to systematically resum these kinds of diagrams in general, but the numerical values of these

higher loop diagrams might be significantly smaller than the ones already considered [28].

H.4 Numerical methods

Numerically, it is easier to solve Eq. (8.23) keeping the IR divergent term explicit.

∫ dk′ dk′ f(ω, k′ ) c (k′ − k ) 2 0 y | ′ | 0 b 0 0 e ky ′ ′ 2 | ′ |3 2 2 2 − 2 β(k −k0) (2π) ( ky + m ) + cb (k0 k0) sinh 0 ∫ ( 2 ) 4 ′ − 1/3 − − 1/3 ′ e dk0dk10 k10 ( ik10) (i(k10 ω)) f(ω, k0) + √ ′ 4/3 − 4/3 − 1/3 − k −k k −k10 24π 3c 2π ( ik10) (i(k10 ω)) (2k10 ω) cosh 0 10 cosh 0 [ b( ( ) ( )) 2T ] 2T ik + πT i(ω − k ) + πT = c T 2/3 H − 0 + H − 0 + 2µ(T ) f(ω, k ). (H.8) f 1/3 2πT 1/3 2πT 0

2 ≪ 2/3 ≪ ′ We keep m finite but small, such that m T and m T . The integration over ky is done

′ numerically. The integration over k0 then is discretized as a matrix multiplication, and the equation

313 0.6 0.030 0.5 0.025 0.4 0.020 0.3 0.015

0.2 0.010

0.1 0.005

0.0 0.000 0 1 2 3 4 5 6 0 1000 2000 3000 4000 5000 6000

3.0 0.10 2.5

2.0 0.05

1.5 0.00

1.0 - 0.05

0.5 - 0.10 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 - 0.03 - 0.02 - 0.01 0.00 0.01 0.02 0.03

Figure H.3: (a) Plot of the magnitude of the smallest eigenvalue for ω on the positive imaginary axis for T = 1.0. (b) Plot of the magnitude of the entries of the corresponding eigenvector when −iω = λL. (c) Plot of λL vs T . (d) Plot 2 of Im[δλL] vs Npx. The value of Re[δλL] ∼ (Npx) is very small when Npx is small. This real part does not control the speed vB⊥ at which the wave pulse of Eq. (8.25) travels, but will lead to the broadening of the pulse as it travels (see below). For all these figures, k0 ∈ [−15, 15], m = 0.02, the step size dk0 = 0.005 and e = 1.0.

is brought to a form M(ω)f(ω) = 0. For a given ω on the positive imaginary axis, we find the eigenvalue of M with the smallest magnitude, which is easier to do than diagonalizing the entire matrix. We then use the Newton-Raphson method to find values of ω on the positive imaginary axis for which the smallest eigenvalue of M is zero or nearly zero within a small tolerance. A plot of the magnitude of the smallest eigenvalue as a function of −iω is shown in Fig. H.3(a). We see

that there is one zero for −iω > 0, which gives the value of λL. The corresponding eigenvector is shown in Fig. H.3(b). A plot of λL vs T is shown in Fig. H.3(c).

For the butterfly velocity, we solve Eq. (8.24) using the same technique as in the above. Now λL

314 is no longer purely real when Np ≠ 0, and we numerically find δλL for small Np using the slope x δNpx x of Fig. H.3(d), leading to the result in the main text. In order to determine the function g(t, Npx)

that controls the shape of the wave pulse in Eq. (8.25) of the main text, we need to numerically find

4/3 2/3 δλL to higher orders in Npx. Up to second order we obtain δλL/T ≈ −4.10(iNpx/(e T )) −

2 2 8/3 4/3 2.74(N px/(e T )). This gives

∫ − 2 − (x vB⊥t) 2 10/3 − f 2 1 f N v D⊥pxt λLt 4D t f F g(t, Npx) ∼ e , dy f(t, x) ∼ √ e e ⊥ ,D⊥ ≈ 2.74 (H.9) f e8/3T 1/3 γ2/3 tD⊥

f when factors of vF and γ are restored. The quantity D⊥ has the dimensions and scaling of a diffusion

E constant such as D⊥. However, we are unable to make any comments as to whether any special

E f relation exists between D⊥ and D⊥.

H.5 Specific heat and thermal conductivity

The expression for the free energy may be rewritten as a contour integral, keeping in mind the branch cuts in the fermion propagators along the real frequency axis

∫ ∫ N ∞ dz d2k −1 + − −1 − F = z/T 2 (ln G (z , k) ln G (z , k)) 2πi −∞ e + 1 (2π) ic˜ −1 2 ∓ f ∓ 2/3 G (z , k) = kx + ky ( iz) , ∫ ∫ N ∫ ( ) N dk ∞ dz dk c˜ |z|2/3/(2N) = − y x tan−1 √f z/T 2/3 π 2π −∞ e + 1 2π kx − (˜cf 3/(2N))sgn(z)|z| √ ∫ ∫ ∫ 2/3 ∞ (˜cf 3/(2N))sgn(z)|z| − dky dz dkx N z/T , (H.10) 2π −∞ e + 1 −Λ 2π

315 → − 2 where we shifted kx kx ky to eliminitate ky from the integral and Λ is some large cutoff.

−1 The kx integral over the tan vanishes. The terms that end up contributing to the specific heat

2 2 CV = −T ∂ F/∂T give,

∫ ∫ √ ∞ 2/3 − dky dz z F = c˜f 3 z/T . (H.11) 2π 0 2π e + 1

Evaluating this integral and differentiating with respect to T gives the expression for CV in the main text.

We now turn to the computation of the energy current correlator required to determine κ¯⊥. The contribution which includes the resummation of the one-loop self energy corrections is

∫ ( ) d2k ∑ q 2 ⟨J EJ E⟩(iq ) = N T G˜(k)G˜(k + q ) k + 0 ⊥ ⊥ 0 (2π)2 0 0 2 ( )k0 ∫ 2 2 ∑ q0 | − | N dky k0 + 2 Θ(k0) Θ(k0 + q0) = T 2/3 2/3 2˜cf 2π |k0| + |k0 + q0| k0 ( ) | | 2 ∫ −πT q0 2 ∑ k0 + N dky 2 = T 2/3 2/3 . (H.12) c˜f 2π (−k0) + (k0 + |q0|) k0={−|q0|}

Where by {−|q0|} we mean the first fermionic Matsubara frequency above the bosonic Matsubara frequency −|q0|. The sum can be converted into a (suitably regularized) contour integral

∫ [ ∫ N 2T 7/3 dk ∞ dz 1 ⟨ E E⟩ y J⊥ J⊥ (iq0) = z c˜f 2π 0 2πi e + 1  ( ) ( )  | | 2 | | 2 − q0 q0  iz + 2T iz + 2T  ×  ( ) − ( )  | | 2/3 | | 2/3 2/3 q0 − − 2/3 q0 (iz) + T iz ( iz) + T + iz

316  ( ) ( )  { 2 2 ∫ |q | |q | 0 − 0 0 dz 1  iz + 2T iz + 2T  +  ( ) − ( )  2πi ez + 1 | | 2/3 | | 2/3 −∞ 2/3 q0 − − 2/3 q0 (iz) + T iz ( iz) + T + iz ( ( ) ) }] i z(−iz)1/3 + z(iz)1/3 q2 4iz |q | − 0 − 0 . (H.13) 9 ((−iz)2/3 + (iz)2/3)3 T 2 3 ((−iz)2/3 + (iz)2/3) T

These integrals must be done numerically, and it is then easily seen that they reproduce the sum correctly at bosonic Matsubara q0. When q0 → 0, we find that

∫ 2 4/3 E E N T dky ⟨J⊥ J⊥ ⟩(iq0) ≈ −0.28 |q0| , (H.14) c˜f 2π

which yields the result in the main text after analytic continuation. For the conductivity α⊥, we

have the charge current

∫ ∫ 3 3 d k ∂ϵk † d k † ⊥ J (iq0) = 3 ψ (k + q0)ψ(k) = 3 ψ (k + q0)ψ(k). (H.15) (2π) ∂kx (2π)

Then

∫ 2 ∑ ( ) E d k q0 ⟨J J⊥⟩(iq ) = iN T G˜(k)G˜(k + q ) k + ⊥ 0 (2π)2 0 0 2 k ∫ ( ) 0 2 ∑ q0 | − | N dky k0 + 2 Θ(k0) Θ(k0 + q0) = T 2/3 2/3 = 0, (q0 = 2nqπT ) (H.16) 2˜cf 2π |k0| + |k0 + q0| k0

and hence α⊥ vanishes in our approximation. The momentum integrals in the simple two-loop

E E vertex correction to ⟨J⊥ J⊥ ⟩ were considered in Ref. [28] for the two-loop renormalizations of the

boson propagator. They found that the momentum integrals in the vertex correction vanish, owing

to the obtainment of terms with denominators posessing poles on the same side of the real axis.

317 I Appendices to Chapter 9

I.1 Outline of Feynman rules for the complex-time contour

In this appendix we briefly outline the Feynman rules on the complex-time contour that are used to compute Eq. (9.13). A detailed derivation of Feynman rules for such scenarios has been presented earlier in Refs. [50, 52]. We split the Hamiltonian H into three pieces corresponding to the clean, non-interacting system, the disordered potential, and the interaction term

≡ clean dis H = H0 + Hint Hfree + Hfree + Hint (I.1)

318 clean For Hfree , Eq. (9.13) simply factorizes by Wick’s theorem into a product of a retarded Green’s

function and an advanced Green’s function. When disorder is included, we have

∫ − t dis clean dis T i 0 dtHfree[t,ψfree ] clean ψfree(t, x) =( e )ψfree (t, x) ∫ −i t dtHdis [t,ψclean] † × (T e 0 free free ) , (I.2)

T dis where denotes time-ordering. The exponentials containing Hfree may now be expanded, this pro-

clean duces corrections to Eq. (9.13) with H = Hfree that can be contracted by Wick’s theorem and the

disorder average Eq. (9.5). Since the disorder is time-independent, this produces to lowest order the

disorder self-energy corrections to the Green’s functions (Figure 9.2a), and also the disorder ladder

corrections in Figure 9.3a. These corrections can then be resummed to obtain the non-interacting

f(t, x) as shown in Appendix I.2.

With the inclusion of interactions, we use

∫ − t dis clean clean T i 0 dt(Hfree[t,ψfree ]+Hint[t,ψfree ]) clean ψ(t, x) = ( e )ψfree (t, x) ∫ −i t dt(Hdis [t,ψclean]+H [t,ψclean]) † × (T e 0 free free int free ) . (I.3)

It is helpful to consider for the purposes of this illustration the decoupling the four-Fermion interactions using a bosonic field φ(ω, k) with a propagator given by the unscreened Coulomb interaction

Vb(k). The perturbative expansion now generates the corrections shown in Figure 9.4 that involve

the usual correction to the Green’s functions due to interactions, along with a new set of corrections

319 that involve the contraction of the boson field across the e−βH/2 thermal factors of Eq. (9.13),

V W (ω, k) ≡ Tr[e−βH/2φ(ω, k)e−βH/2φ(−ω, −k)]. (I.4)

The expression for this Wightman propagator V W is provided in Eq. (9.21), and its relation to

the spectral function is derived in detail in Refs. [52, 286]. These new corrections generate the

diagrams shown in Figure 9.5. Note that interaction corrections to the e−βH/2 thermal factors in

Eq. (9.13) correspond to the dressing of the Wightman propagators, which we take into account

since we use the dynamically screened Coulomb interaction for V W in Eq. (9.21).

I.2 Absence of chaos in the non-interacting disordered metal

In this appendix we derive the expression for f(t, x) in the non-interacting scenario. We have (see

Eq. (9.13) and Figure 9.3),

∫ ddkdk f(ω, q) = 0 GR(k + ω, k + q)GA(k , k)+ (2π)d+1 0 0 0 0 ∫ ddk ddk dk 1 2 0 GR(k + ω, k + q)GA(k , k )GR(k + ω, k + q)GA(k , k )L(ω, q). (I.5) (2π)2d 2π 0 0 1 0 0 1 0 0 2 0 0 2

The diffuson rung L(ω, q) is given by the following resummation of disorder rungs:

∫ ddk L(ω, q) = U 2 + U 2 GR(k + ω, k + q)GA(k , k)L(ω, q) 0 0 (2π)d 0 0 0 0 ∫ ddk 1 1 = U 2 + U 2 L(ω, q) 0 0 (2π)d k2 − µ − k + i (k+q)2 − − − − i 2m 0 2τ 2m µ k0 ω 2τ

320 ∫ ddk 1 1 ≈ U 2 + U 2 0 0 d k2 i k2 i (2π) − µ − k0 + − µ − k0 −  2m ( 2τ 2m ) 2τ 2 ω k · q/m × 1 + +  L(ω, q) k2 − − − i k2 − − − i 2m µ k0 2τ 2m µ k0 2τ ∫ ( ) dϵ 1 ω q2v2 /d ≈ U 2 + U 2g(0) 1 + + ( F ) L(ω, q). (I.6) 0 0 − 2 1 − − i i 2 2π (ϵ k0) + 2 ϵ k0 − − 4τ 2τ ϵ k0 2τ 1 L(ω, q) = . (I.7) g(0)τ 2(−iω + Dq2)

2 where D = vF τ/d, and in the intermediate steps, we expanded in small q assuming that the largest contributions to the integrals come from the regions with k ∼ kF = mvF ≫ q, and that µ ≫

τ −1 ≫ |ω|. We assumed that L(ω, q) does not depend on any other combinations of momenta and frequencies passing through it apart from (ω, q), which turns out to be self-consistent. Each disorder rung is multipled by a factor of −i2 = 1, where the i’s come from the real-time electron-

−x2/(4Dt) disorder vertices. We thus see that f(t, x) ∼ f0(t, x) + f1(t, x)e , where f0 decays rapidly

−1 in time at a rate given by τ and f1 is a slowly varying function of space and time. Henceforth we ignore f0 as we are interested in long times t ≫ τ and set f1 to 1. Since there is no exponential growth in f(t, x) we conclude that the non-interacting disordered metal does not have many body quantum chaos.

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