Developing Dynamical Probes of Quantum Spin Liquids Inspired by Techniques from Spintronics

City University of New York (CUNY) CUNY Academic Works

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9-2021

Joshua Aftergood The Graduate Center, City University of New York

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This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Developing dynamical probes of quantum spin liquids inspired by techniques from spintronics

Joshua Aftergood

A dissertation submitted to the Graduate Faculty in Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy, The City University of New York

2021 ii

c 2021

Developing dynamical probes of quantum spin liquids inspired by techniques from spintronics by Joshua Aftergood

This manuscript has been read and accepted by the Graduate Faculty in Physics in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy.

Date Professor So Takei Chair of Examining Committee

Date Professor Alexios Polychronakos Executive Oﬃcer

Supervisory Committee: Professor Alexander Lisyansky Professor Azriel Genack Professor Eugene Chudnovsky Professor Branislav Nikolic´

The City University of New York iv

Abstract Developing dynamical probes of quantum spin liquids inspired by techniques from spintronics by Joshua Aftergood

Advisor: Professor So Takei

We theoretically study low dimensional insulating spin systems using spin fluctuations as a probe of the spin dynamics. In some systems, low dimensionality in conjunction with other quantum fluctuation enhancing effects impedes spontaneous generation of long range magnetic ordering down to zero absolute temperature. In particular, we focus on exotic spin systems hosting mobile, fractionalized spin excitations above their ground state, and ultimately show that techniques already commonplace in spintronics can be utilized in the context of quantum magnetism to develop probes of exotic ground states. We initially consider quantum spin chains (QSCs), and examine a system of two exchange- coupled QSCs connected at their finite ends. Downstream, one QSC is driven out of equilibrium by both an overpopulation of spin excitations and temperature elevation, and the other QSC is coupled to a drain bath where electrical measurements are performed via spin Hall effects. We compare this low dimensional system hosting spin-half solitons to a coupled 3d magnon hosting system arranged in the same geometry and show that a quantity we call the spin Fano factor, defined as the spin current noise-to-signal ratio, differentiates between the two systems. However, we assume that the noise is measured electrically via the inverse spin Hall effect after conversion in the drain metal. Recognizing that this technique may lead to enhancements due to intrinsic conversion mechanisms [1], we then suggest an experimental method by which to measure spin current noise minimally invasively using an LC resonator coupled to a transmission line. In this way, we show that photon counting allows for directly accessing the spin current dc noise in a v

QSC, and thus for constructing the spin Fano factor. Moving on to 2d insulating magnets, we examine a bilayer system interfacing a spin-orbit coupled heavy metal film with an exotic quantum magnet. We develop the theoretical underpinnings of the proposed bilayer system for a general quantum magnet lacking long-range magnetic order, and then apply the theory to three QSL models. We select the Heisenberg kagome´ lattice model, a model possessing gapless fermionic spinons and an emergent U(1) gauge field, and the Kitaev model on the honeycomb lattice, each with extant candidate materials in mind. Ultimately, a bilayer system of the type proposed can utilize interfacial spin fluctuations as a probe of the low energy density of states of QSL materials, and access quantities of interest in these systems such as gap energies and gauge field renormalizations. Additionally, in any multilayer system, the interface is of obvious interest, and so we close with a microscopic examination of a metal-to-quantum magnet interface. The metal contains short-range, spin-inert impurities, the insulator provides a bath of spin fluctuations, and the interface itself breaks inversion symmetry and results in interfacial Rashba spin-orbit coupling. We calculate the electrical conductivity in the metallic layer in the presence of all three of these factors, and show that corrections to the conductivity may themselves be utilized as a probe of the affixed insulating spin system. Acknowledgments

The completion of this thesis was made possible by the support of many people both inside and outside of academia. First, I would like to gratefully acknowledge my advisor, So Takei. It was his passion and dedication that initially brought me into condensed matter physics, and I will always be thankful for that. His subsequent patience and guidance were instrumental to my work over the last ﬁve years, and central to the conclusion of this undertaking. I would also like to sincerely acknowledge my other committee members: Alexander Lisyan- sky, Azriel Genack, Eugene Chudnovsky, and Branislav Nikolic´ for their willingness to oﬀer their time and energy to this laborious process. My thanks as well to Subrata Chakraborty for our discussions, both relating to physics and not. Finally, I would like to acknowledge the enormous help of my family. My wife Mattea has been of inestimable value in maintaining my sanity throughout the process, and central to my ability to continue working in spite of the craziness of this past year. Pax and Aurelia — in spite of their boundless energy and ability to break anything — helped me regain motivation in the inevitable slumps, though they are too young to understand.

vi Contents

Contents vii

List of Figures xi

1 Introduction 1

1.1 Quantum magnets ...... 1 1.2 Spin ﬂuctuations: spintronics ...... 4 1.3 Overview of the Thesis ...... 7

2 Necessary formalism 9

2.1 The closed time contour ...... 10 2.1.1 Adiabatic switching ...... 12 2.1.2 The equilibrium limit ...... 14 2.2 Keldysh Green functions ...... 15 2.2.1 The Kadanoﬀ-Baym basis ...... 18 2.2.2 The Keldysh basis ...... 19 2.2.3 The ﬂuctuation dissipation theorem ...... 21

3 Spin shot noise across coupled 1d spin chains 22

3.1 Introduction ...... 23

vii CONTENTS viii

3.2 Model ...... 25 3.2.1 Luttinger model for the spin chains ...... 26 3.2.2 Exact mapping to the problem of stochastic electron tunneling between two fractional quantum Hall edge channels ...... 27 3.3 Spin current and dc noise ...... 30 3.3.1 Spin current ...... 31 3.3.2 Dc noise ...... 33 3.3.3 Spin transport between two magnon baths ...... 36 3.4 Discussion ...... 39 3.4.1 Spin Fano factor ...... 39 3.4.2 Manifestation of Pauli blockade in noise ...... 41 3.5 Experimental extraction of spin Fano factor ...... 43 3.5.1 QSC scenario ...... 43 3.5.2 Magnon scenario ...... 46 3.5.3 Estimate of noise signal strength ...... 48 3.6 Conclusion ...... 49

4 Measuring spin shot noise with an LC resonator 51

4.1 Introduction ...... 52 4.2 Heuristic picture ...... 54 4.3 Microscopic theory ...... 56 4.3.1 The Model ...... 56 4.3.2 Input-Output Formalism ...... 58 4.3.3 The Output Photon Flux Spectrum ...... 59 4.4 Noise detection ...... 61 4.4.1 AC Noise Detection ...... 61 CONTENTS ix

4.4.2 The DC Limit and the Fano Factor ...... 63 4.5 Conclusion ...... 64

5 Measuring QSL density of states using spin ﬂuctuations 65

5.1 Introduction ...... 66 5.2 Theory ...... 69 5.3 Application to Quantum Spin Liquids ...... 73 5.3.1 Heisenberg antiferromagnet on the kagome´ lattice ...... 73 5.3.2 Spinon Fermi surface coupled to an emergent U(1) gauge ﬁeld ...... 79 5.3.3 The Kitaev Honeycomb Model ...... 90 5.4 Physical Estimates ...... 98 5.5 Discussion and Conclusions ...... 100

6 Paramagneto conductance and bilayer interfaces 105

6.1 Introduction ...... 106 6.2 The Model ...... 108 6.3 Linear response and the Kubo formalism ...... 110 6.4 Contributing diagrams ...... 112 6.4.1 The self-energy terms ...... 113 6.4.2 The vertex correction terms ...... 117 6.4.3 The barbell terms ...... 119 6.4.4 The Hartree diagrams ...... 120 6.5 Disorder averaging ...... 122 6.6 Diamagnetic cancelation and a well behaved dc limit ...... 128 6.6.1 Diamagnetic diagrams ...... 129 6.7 The Conductivity ...... 132 CONTENTS x

6.7.1 Self-Energy and Vertex Contributions ...... 134 6.7.2 The Z 0 Limit ...... 147 → 6.7.3 The conductivity ...... 149 6.8 Discussion and conclusions ...... 152

7 Conclusion 159

Bibliography 162 List of Figures

1.1 Diagram a) depicts the geometric frustration of an antiferromagnetic (J > 0) triangular lattice. The arrows represent spin directions. If the bottom two spins are arrayed in the standard antiferromagnetic up-down conﬁguration, the third spin is unable to minimize its exchange energy with respect to the other two spins — it is geometrically frustrated. Diagram b) depicts exchange frustration. These spins are coupled with a ferromagnetic Ising interaction in the x, y, and z directions, such that the central spin is unable to simultaneously minimize its energy with respect to all three directions. It is then experiencing exchange frustration. This ﬁgure is taken from Ref. [2]...... 3

2.1 Diagram (a) depicts the contour presented by Eq. (2.6), where evolution begins

in equilibrium at time t0, progresses forward until time t where the operator is evaluated, and then back to equilibrium. Figure (b) is a cartoon of the contour, denoted c, after extension to t = by insertion of the unitary evolution operators. ∞

Image (c) depicts the Keldysh contour, cK, where evolution proceeds from the infinite past to the infinite future and back, attained after invoking the adiabatic assumption...... 13 2.2 The different time-ordering configurations possible on the Keldysh contour. Red

dots represent ﬁeld operators at the depicted times, t or t0, on a branch of the contour. 18

xi LIST OF FIGURES xii

3.1 Two semi-inﬁnite quantum spin chains, one labeled the source and the other the drain, are coupled at their ﬁnite ends. Strongly spin-orbit coupled metals act as

injector and drain for spin current. The source chain is held at temperature T1

and the drain chain at T2. The spin injection from the injector metal via the spin Hall effect leads to a non-equilibrium accumulation of spin in the source chain. Depicted positions are continuum variables...... 24 3.2 The problem of tunneling spin-1 excitations between two semi-infinite QSCs exactly maps to the problem of stochastic electron tunneling between the edge states of two Laughlin fractional quantum Hall liquids at filling fraction K. The role of the spin chemical bias in the QSC problem is played by the electrical voltage µ = eV applied to the edge channel on the left in the quantum Hall case...... 30 3.3 Depiction of proposed magnonic system. Two coupled magnon baths are held at

temperatures T1 and T2 and chemical potentials µ1 and µ2, respectively...... 36 3.4 Spin Fano factor in the QSC and magnon cases plotted as a function of the chemical bias µ µ . Here, τ = (T T )/T , and we set µ = 0 and T T in both systems. 40 ≡ 1 1 − 0 0 2 2 ≡ 0

3.5 Excess dc spin current noise as a function of µ2/µ1 for the QSC case (solid line)

and the magnon case (dashed line) with T1 = T2 = T0. See the main text for the

deﬁnitions of α(µ1, µ2) and αm(µ1, µ2). The suppression obtained in the QSC case is a manifestation of Pauli blockade physics...... 43 3.6 Depiction of the proposed conﬁguration for extracting the spin Fano factor in the QSC case. Two source chains are individually coupled to a third drain chain at

one point. The chains are held at temperatures T1, T2, and T0 respectively, and

chemical biases µ1, µ2 in the source chains only...... 44 LIST OF FIGURES xiii

3.7 Depiction of the proposed conﬁguration for extracting the spin Fano factor in the magnon case. Two source baths are individually coupled to a third drain bath. The

baths are held at temperatures T1, T2, and T0 respectively, and chemical biases µ1,

µ2 in the source baths alone...... 46 3.8 Depiction of spin to charge current conversion between a QSC and normal metal contact. We extract an approximation for the size of the conversion eﬀect between the two materials from this geometry. Here, l is the thickness of the interface and d is the width of the metal contact...... 48

4.1 A depiction of the proposed system. A microwave resonator, comprised of a loop of inductance L and a capacitor of capacitance C, is set in the xy plane a distance

δ from spin chain 2, which is coupled with strength ξc to spin chain 1...... 53 4.2 A plot depicting magnetic ﬂux through the loop as spins enter spin chain 2. The

sharp drop-offs indicate that spins outside of the “influence region” x1 < x < x2 are effectively ignored by the microwave resonator...... 55

4.3 A plot of N(Ω) as a function of transmission line temperature Tt. Here, we use Ω/2π = 1 GHz and spin temperature T = 4 K to remain in the dc spin transport

regime, and spin bias µ = 0.03J. The plateau in the limit of small Tt (shaded in blue) is the dc spin current noise. N(Ω) exhibits the same qualitative behavior for essentially all spin biases µ, thus allowing one to quantify the noise using this extraction method for various values of µ...... 62

5.1 A cartoon of the proposed system: a quantum spin liquid to normal metal bilayer

in equilibrium, where ﬂuctuating spin current Is across the interface becomes ﬂuc-

tuating charge current Ic (or fluctuating voltage) in the normal metal via the inverse spin Hall effect and results in measurable modifications to the ac resistance of the metal...... 70 LIST OF FIGURES xiv

5.2 The kagome´ lattice, with positions on a representative unit cell marked. The eˆi represent the unit vectors of a unit cell, and a is the length of a bond. Arrows between

sites correspond to the direction dependent Qi j. Solid arrows around upward fac-

ing triangles depict Qi j = Q1, and wire arrows around downward facing triangles

depict Qi j = Q2...... 75

5.3 A numerical plot showing S s(Ω, 0) in the HKLM as a function of ac frequency normalized by the mean-ﬁeld coupling, λ = 0.695J. The main plot shows the computed sum. The graph is identically zero until a critical frequency of twice the

gap energy is achieved, ~Ωc = 2∆s = 0.158λ, at which point it becomes possible to create spinon pairs in the QSL and noise enhancement commences. The inset is the ac susceptibility χ(Ω, 0) in the case of the HKLM, which can be extracted by numerical differentiation of Eq. (5.8), and is in good agreement with previous works, see, e.g., Ref. [3]...... 78 5.4 Three diagrams contributing to the (1/ ) correction to the spin susceptibility O N generated by the presence of U(1) gauge field fluctuations...... 89 5.5 The Kitaev honeycomb lattice, where the two sub-lattices are shown by the purple (sub-lattice A) and orange (sub-lattice B) dots. Representative x-, y-, and z-links

are depicted. The ﬂux operator on plaquette p, Wp, is shown, n1 and n2 are the primitive lattice vectors, and a is the lattice spacing. Also depicted is a representative unit cell enclosed in the dashed box...... 92 LIST OF FIGURES xv

5.6 A numerical plot showing S s(Ω, 0) for the Kitaev model as a function of ac frequency normalized by the isotropic interaction strength, K. The main plot shows the computed sum. The graph is identically zero until a critical frequency equal to the gap energy is achieved, ~Ω = ∆ 0.26K, at which point it becomes pos- c F ≈ sible to inject ﬂux pairs into the QSL and noise enhancement commences. Here, N = 4 104 has been used. The inset is the ac susceptibility χ(Ω, 0) in the case of u × the Kitaev model, which can be extracted via numerical diﬀerentiation of Eq. (5.8), and is in good agreement with previous works [4]...... 98

6.1 A cartoon of the proposed bilayer system. An insulating magnet, e.g., a quantum spin liquid, is coupled to a weakly spin-orbit coupled metal thin ﬁlm. The interface breaks inversion symmetry causing Rashba spin orbit coupling in the metal

layer. As a result, the current in the x direction, jc, carries signatures from both the interfacial spin orbit coupling and proximity to the insulating magnet...... 108 6.2 The six possible diagrams, derived from Eq. (6.13). Solid lines depict fermion propagators, and dashed lines spin propagators. Diagrams a) and b) are the self- energy corrections, diagram c) is the vertex correction, d) is the barbell diagram, and e) and f) are the Hartree diagrams...... 112 6.3 The two self-energy correction diagrams with energy ﬂow around the bubble depicted. Solid lines are fermion propagators, and dashed lines are spin propagators. The real space coordinates are listed, for ease of calculation. Numbers associated with each fermion line correspond to those listed in the Green functions in the main text...... 114 LIST OF FIGURES xvi

6.4 The diagram for the vertex correction with energy flow and real-space positions noted. Solid lines are fermion propagators, and the dashed line represents a spin propagator. Numbers associated with each fermion line correspond to those listed in the Green functions in the main text...... 117 6.5 The barbell diagram with energy flow and real space positions noted. Solid lines are fermion propagators, and the dashed line represents a spin propagator. Num- bers associated with each fermion line correspond to those listed in the Green function arguments in the main text...... 119 6.6 The Hartree diagrams with energy flow and real space positions noted. Solid lines are fermion propagators, and the dashed line represents a spin propagator. Num- bers associated with each fermion line correspond to those listed in the Green function arguments in the main text...... 121 6.7 A pictorial depiction of the current vertex correction. The filled in dot is the corrected current vertex, and the empty dot is the uncorrected vertex. Impurity scattering events are denoted by stars and dashed lines. An impurity ladder is inserted in the top line, and then resummed into the Dyson equation as shown in the second line...... 123 6.8 A pictorial rendition of the impurity ladder, where scattering occurs between two Green function lines. Impurities are inserted sequentially, and the resummed equation can be found in Eq. 6.31 of the main text...... 125 6.9 The spin vertex, τα, where α = x, y, z is the spin polarization, dressed by the presence of a ladder. The corrected vertex is Γ...... 127 6.10 Diagrammatic representation of the 2 corrections to the diamagnetic current. Di- J agram a) is the exchange component and diagram b) the Hartree component. . . . . 130 LIST OF FIGURES xvii

6.11 The terms emerging from Eq. eqrefse and Eq. (6.54), for χR, that admit only the insertion of a single cross-bubble ladder. The ladder is highlighted in grey, and energy and momentum ﬂow around the bubble are depicted. Note that inserting the ladder breaks the bubble into two separate momentum sums, one for k and the

other for k0, by accounting for elastic scattering against impurities...... 137 6.12 Diagrams in which it is possible to insert a corrected spin vertex in addition to a cross-bubble ladder. The ladder is depicted in grey, and triangles over spin vertices indicate a corrected vertex...... 140 6.13 Diagrams that can sustain the insertion of two corrected spin vertices in addition to a cross-bubble ladder. The cross-bubble ladder is depicted in grey, and the corrected spin vertices appear as triangles around spin interactions. Cross-bubble ladders break the bubble up into two halves, each with its own momentum, eﬀectively decohering the charge vertices...... 143

6.14 Two additional diagrams arise out of the RARAa, and similarly RARAb, term due to the ability to connect the two retarded and two advanced Green functions with a single impurity scattering event, denoted by the star connected to a dotted line. These situations are possible in this diagram and not the others due to the fact that this particular arrangement of Green functions maintains two separate non-zero sums after a scattering event...... 146 6.15 The two zeroth order terms. Diagram a) is the bare current-current bubble, and diagram b) is the diamagnetic term associated with the bare bubble...... 150 Chapter 1

Introduction

1.1 Quantum magnets

Quantum magnets refer to a panoply of interacting spin systems that exhibit a rich array of quantum phases and phase transitions. These quantum many-body systems can be broadly classified into two distinct categories: what we may term conventional and unconventional quantum magnets. Conventional quantum magnets exhibit traditional long range magnetic ordering along either ferro- or antiferromagnetic lines, and their ground states are essentially classical spin configurations with quantum effects entering perturbatively. These conventional phases support low-energy collective spin-one (in units of ~) charge neutral excitations known as magnons — quantized har- monic fluctuations of the ordered spins. Magnons are mobile, and therefore conventional magnetic phases supporting magnons are able to propagate spin transport quantities regardless of the fact that quantum magnets are electrically insulating. The physical properties of these magnons, such as their mode dispersion and response functions, can be calculated using well-established quantitative theories like the spin wave theory, and it is in this sense that conventional quantum magnets are conventional. Unconventional quantum magnets, by contrast, exhibit no magnetic ordering down to temper-

1 CHAPTER 1. INTRODUCTION 2 atures much lower than their interaction scales — in fact, they often lack magnetic ordering into the millikelvin range. A prototypical example of this family of quantum magnets is the quantum spin liquid (QSL). QSLs are an intriguing state of matter in which quantum fluctuations are strong enough to melt magnetic ordering all the way down to zero temperature. These QSLs are distinguished among themselves by different quantum orders characterizing their extensive many-body quantum entanglement as opposed to which space-time or spin rotational symmetries they break. Quantum non-locality is therefore the key feature of QSLs, and results in an array of unusual and interesting phenomena: topological properties, non-trivial correlations, and exotic excitations with fractionalized quantum numbers and statistics, to name a few [5; 6]. The centrality of quantum mechanics to understanding QSLs makes beginning from a classical configuration and introducing quantum effects perturbatively impossible — rather, quantum mechanics forms the core of approaches to modeling QSL phases [7; 8; 9]. Nevertheless, the theoretical understanding of QSL phases is far from complete, and remains a topic at the frontier of condensed matter physics. An important implication of quantum non-locality in the QSL phase is a QSL’s ability to support non-local excitations. Non-local excitations, e.g., solitons and spinons, are excitations that cannot be created by any individual local operator; instead, they emerge as the result of an infinite product of local operators, i.e., the collective behavior of the system. A key example of these non- local excitations are gapless solitons arising in the spin-half Heisenberg antiferromagnetic chain (HAC). These solitons carry spin-half [10] in spite of the fact that each site of the HAC is a spin- half operator, in which case a spin flip corresponds to a complete spin-one insertion. The HAC has received extensive research attention due to its amenability to rigorous theoretical analysis [11] and the fact that well-known material options exist [12]. It is the simplest and best established example of an insulating magnet possessing a QSL ground state due to reduced dimensionality. In systems beyond one dimension, however, the existence of a true QSL ground state remains controversial [13]. A great deal of research, both experimental and theoretical, has revolved around the search for CHAPTER 1. INTRODUCTION 3 a) b)

(a) Geometric frustration (b) Exchange frustration Figure 1.1: Diagram a) depicts the geometric frustration of an antiferromagnetic (J > 0) triangular lattice. The arrowsFigure represent 1.2.1.: (a) The spin geometrically directions. frustrated If the bottom triangular two lattice spins with areantiferromagnetic arrayed in the in- standard antiferromagnetic up-downteraction configuration, (J>0). The the arrows third indicates spin is unablethe spin direction. to minimize If two neighboringits exchange energy spins are antiferromagnetically ordered, the exchange energy between the with respect to the other twothird spins spin and — both it is its geometrically neighbors can not frustrated. be simultaneously Diagram minimized.b) depicts This exchange frustration. These spins arespin, coupled indicated with by a a question ferromagnetic mark, is then Ising frustrated. interaction (b) The in spinsthe x couple, y, and z directions, such that the centralwith spin a isferromagnetic unable toIsing simultaneously type interaction minimize in x, y, z-direction its energy with with its neigh- respect to all three directions. It is thenbors. experiencing Since the central exchange spin can frustration. not be parallel This to figure the x, y, is z-axis taken simultane- from Ref. [2]. ously, it is frustrated due to its exchange interaction. Figure taken from Ref. [6]. an uncontroversial material instantiation of a 2d QSL phase. The search has focused primarily on and exotic phenomena with fundamental theoretical implications. Nevertheless generations spin-half quantumof physicist antiferromagnets have not yet succeeded with highly in solving frustrated seemingly lattices. simple Figure models. 1.1 illustrates that frustration comesSo in how two do forms: we find geometric fractional frustration,excitations and which other occurs exotic behavior?when the spinsAs mentioned, are arrayed mag- in such netic interaction often lead to an ordered ground state and only when thermal fluctuations a manner thatare strong certain enough spins in the the order lattice is broken are unable and a toparamagnetic minimize their state energy emerges. by This adopting happens a particular configurationabove a critical with respect temperature to theTc. others, However and there exchange can be fluctuations frustration, of in a completely which case different it becomes origin, that lead to the suppression or destruction of order. This is the case when different impossibleforces for spins compete to simultaneously with each other. satisfyIf they cannot all of be the minimized spin-dependent simultaneously, bond interactions the system is acting called frustrated. upon it. In either case, frustration results in highly degenerate ground states in the corresponding A simple example is the geometrically frustrated triangular lattice, see Fig. 1.1(a). J>0 classical spinindicates models, antiferromagnetic and appears to interaction, be a necessary which means component neighboring of a stable spins QSL favor phase. anti-parallel Other sta- alignment. If, however, two spins on the edges of the triangle are already orientated in this bilizing influencesmanner, there can be is no any possibility factors tothat place enhance the third quantum one in such fluctuations: a way, that reduced it is anti-parallel dimensionality, to small spinboth sizes, its neighbors.and small For coordinate the small numbers. triangle this Though already leads controversy to six degenerate remains, ground several states. promising In this example the spins are strongly correlated, but the competing interactions prevent material candidatesthe formation exist; of a examples stable structure. include They the remain mineral disordered herbertsmithite and do not [14], shown triangular any symmetry lattice an- breaking. tiferromagnet materials [15], and Kitaev compounds such as H3LiIr2O6 [16]. These materials have been the subject of a wide variety of experimental studies, but no clear signal of a true QSL phase has yet been demonstrated. One clear signal that could4 be presented is to demonstrate the presence of the mobile, non-local, fractionalized excitations that are thought to exist in the QSL phase — CHAPTER 1. INTRODUCTION 4 spin-half solitons in the HAC, or spinons in 2d QSLs. New methods of probing candidate QSL materials may help in the search for a true QSL ground state, and this thesis will propose spin fluctuations as a viable probe of QSL candidate materials that can shed light on this intriguing state of matter in both one and two dimensions, and potentially discover a signal of mobile spinons.

1.2 Spin ﬂuctuations: spintronics

One of the unifying concepts of this thesis is utilizing spintronics techniques to better understand unconventional quantum magnets. In particular, spin current and its fluctuations will figure prominently throughout. Spin current refers to the net flow of spin angular momentum through various condensed matter media, and spin current fluctuations are the dynamic correlations of the spin current. Unlike charge currents, spin currents can propagate through insulating materials — mediated, for example, by magnons — which makes injecting and detecting spin current a natural method of probing magnetic insulators. The field of spintronics involves the manipulation of spin degrees of freedom, and has thus been intimately tied with research in insulating magnets. Two-terminal spin transport experiments in conventional quantum magnets, for example, have been performed by capitalizing upon spin Hall effects (SHEs) for both pure spin current injection at one terminal and detection at the other [17; 18]. It is only recently that non-equilibrium spin current has been shown to propagate in unconventional quantum magnets hosting non-local, fractionalized spin excitations as well — first in an HAC material [19] and then later in certain 2d QSL systems [20] — which opens up the fascinating possibility of utilizing spin transport quantities to probe these exotic states of matter. Non-equilibrium spin current fluctuations are of particular interest here. This so-called shot noise, by analogy to out-of-equilibrium charge systems, contains information about the effective spin and quantum statistics of the spin-carrying quasiparticles during transport. Charge transport through conductors in the low temperature limit, where thermal fluctuations are suppressed and CHAPTER 1. INTRODUCTION 5 non-equilibrium shot noise dominates, has been a successful experimental tool for detecting the effective charge of, e.g., charge 2e Cooper pairs through superconducting tunnel junctions [21] and fractionalized charge quasiparticles tunneling between edge states in fractional quantum Hall experiments [22]. This thesis will transplant these ideas from charge systems with charge quasiparticle tunneling to insulating magnets with spin quasiparticle tunneling between two systems hosting fractionalized spin excitations, with the idea of better understanding the tunneling spin quasiparticles and their statistics. In keeping with non-local spin transport experiments performed in conventional quantum magnets, the SHE and its inverse will be utilized as spin current injection and detection mechanisms at various points in what comes. In this vein, SHEs become an important building block of this thesis. The SHE, and its inverse, act as transducers between charge and spin current densities in metals possessing strong spin orbit interactions. Spin orbit interactions couple the charge and spin degrees of freedom of electrons, such that momentum flow in one sector induces transverse momentum flow in the other. Thus the direct SHE refers to the generation of a transverse spin current by a charge current, while the inverse SHE is the Onsager reciprocal process whereby an impinging spin current generates a transverse charge current. These phenomena are essential components of the spintronics reper- toire [23], and one of the goals of this thesis will be demonstrating how they can be purposed towards developing probes of unconventional quantum magnets. For instance, interfacing strongly spin-orbit coupled metals (usually 5d transition metals, e.g., Pt, Ta, W) with insulating magnets has been a fruitful arena for studying the magnetization dynamics of the insulating layer — thus far usually consisting of a conventional quantum magnet hosting magnons. Interfacial effects capitalizing upon SHEs underly various phenomena commonplace in spintronics. These include spin pumping [24; 25], where modulating the bulk magnetization of an insulator drives a spin current into a coupled metal, interfacial spin torques, where charge currents driven through the metallic layer can induce a torque on the bulk magnetization of the insulator [26; 27; 28], and spin Hall magnetoresistance, a phenomenon whereby metallic CHAPTER 1. INTRODUCTION 6 conductivity modulates according to the direction of the bulk magnetization in the coupled insulator [29; 30; 31; 32; 33; 34; 35; 36]. Spin Hall magnetoresistance (SMR) in particular is a powerful technique, the utility of which is highlighted by its ability to probe bulk properties like magnetic textures [37; 38; 39; 40; 41; 42] in addition to interfacial properties such as spin-sink and spin-mixing conductances, utilizing e.g., ferri- [32; 33; 34; 35], ferro- [43], and antiferromagnetic insulators [39; 44; 45], that govern interfacial spin opacity in bilayers. These effects, however, tend to involve the bulk magnetic order parameter of the magnetic insulator, and unconventional magnetic insulators lack long range magnetic order — in the case of QSLs in principle all the way to zero temperature — which would seem to preclude bilayers of this type as viable probes of their spin dynamics. Nevertheless, recent experiments utilizing paramagnets in similar bilayer setups have found SMR-like signals in the metallic conductivity [46; 47; 48]. These findings have therefore broad- ened the scope of this bilayer SMR-type physics to include insulators with no long range magnetic order parameter, and indicated that the spin dynamics of insulating paramagnets may be accessible in a bilayer setup via relatively straightforward conductivity measurements. Part of this thesis is therefore purposed toward developing an understanding of magnetotransport in bilayer setups when a spin-orbit coupled metal is interfaced with a QSL material, and how spin fluctuations in- evitably arising at the interface affect charge transport in the metal. Therefore, in addition to non- equilibrium spin shot noise, equilibrium thermal spin fluctuations will themselves prove useful in better understanding quantum paramagnets. Finally, in furthering our understanding of unconventional paramagnetic phases utilizing spin fluctuations and electrical detection methods, the inverse SHE takes on particular importance due to its being the mechanism by which conversion of spin to charge transport quantities occurs. For spin to charge current, the spin Hall angle characterizes the efficiency of this conversion process in strongly spin-orbit coupled metals. In the case of spin to charge current noise, however, the conversion process is not necessarily as straightforward [1]. An examination of that process across CHAPTER 1. INTRODUCTION 7 a metal-to-unconventional quantum paramagnet interface is therefore a useful contribution to a general understanding of these systems in addition to potentially providing insight into the spin dynamics of unconventional quantum paramagnetic phases. In an attempt to tie theory to experiment, this thesis will take the first steps in a microscopic examination of that interface and the conversion of spin-to-charge fluctuations across it.

1.3 Overview of the Thesis

This thesis is organized as follows: Chapter 2 introduces the main theoretical technique used in each subsequent section, namely the Keldysh diagrammatic formalism in the operator description. It begins with a discussion of the closed time-loop contour, which serves as the basis for non- equilibrium quantum field theory. The closed-time contour is then extended to the Keldysh contour via the adiabatic assumption and the subsequent adiabatic introduction of interactions. With the Keldysh contour defined, linearly dependent Green functions describing operator correlations at all points on the contour in the Kadanoff-Baym basis are introduced. Finally, the set of linearly independent Green functions in the Keldysh basis are produced, and explicit functional forms for the case of the non-interacting limit are defined. Chapters 3 and 4 focus on 1d unconventional quantum paramagnets. In Chapter 3, we investigate a system of weakly exchange coupled 1d quantum paramagnets — 1d spin chains — hosting mobile fractionalized spin quasiparticles as their gapless ground state excitations. One spin chain is driven by the injection of a non-equilibrium spin current downstream, and spin shot noise is engendered at the weak coupling between the chains. We then calculate the spin shot noise downstream in the second spin chain. Finally, we compare this system to a similarly configured 3d conventional spin system hosting magnons and introduce a quantity by which to differentiate coupled conventional and unconventional spin systems which we denote the spin Fano factor. Chapter 4 dovetails off of Chapter 3 by introducing an experimental procedure by which to non-invasively CHAPTER 1. INTRODUCTION 8 measure the spin shot noise downstream in the second spin chain using an LC resonator coupled to a transmission line, sidestepping the need for destructive detection mechanisms revolving around the inverse SHE as originally constructed. In Chapters 5 and 6 we shift focus to 2d unconventional quantum paramagnets. In Chapter 5, we investigate a bilayer system that interfaces a strongly spin-orbit coupled metal to various QSL models, and calculate the modification to the resistance across the metal that arises due to the presence of thermal spin fluctuations across the interface. We discover that this type of bilayer setup acts as a probe of the low energy density of states of the QSL system, and show how properties of the spin system — in particular gap energies and signatures of gauge fluctuations — can be extracted from the ac resistance. Chapter 6 continues with the examination of paramagnets in metal-to-paramagnet bilayers with a focus on the physics of the interface itself. We therefore calculate the modification to the conductivity arising in a bilayer affixing a 2d quantum paramagnet to a metal modeled as a 2d electron gas in the presence of spin-neutral disorder, spin fluctuations, and weak interfacial spin-orbit coupling. In so doing, we take the first step in developing a microscopic understanding of the noise conversion process across these interfaces. The material presented in Chapters 3 and 4 are published in Physical Review B [49; 50], and the material presented in Chapter 5 is published in Physical Review Research [51]. The material presented in Chapter 6 is awaiting publication. Chapter 2

Necessary formalism

This chapter is purposed towards introducing the Keldysh diagrammatic formalism, and will serve as the technical foundation for the remainder of this thesis. The Keldysh diagrammatic formalism has the advantages of not requiring the use of analytic continuation, in addition to naturally extending to out-of-equilibrium statistical mechanics, and is used extensively in this thesis as a result. We will apply the formalism in each subsequent chapter to spin systems in and out of equilibrium with the goal of calculating some observable quantity in each case. The main tools we will focus on in that vein are the Green functions, and the ultimate goal of this chapter is the motivation for, definition of, and properties of those Green functions. We will begin by introducing the closed time path formalism and its application to systems out-of-equilibrium; from there, we will define the Green functions on the closed time contour; and finally, we will examine certain properties of those Green functions that we will avail ourselves of in what follows. Details of calculations using the machinery we develop will be relegated to future chapters.

9 CHAPTER 2. FORMALISM 10 2.1 The closed time contour

Consider a quantum many-body system governed by the Hamiltonian

Hˆ = Hˆ 0 + Hˆ i , (2.1)

where Hˆ 0 is quadratic in ﬁelds and represents free particles, and Hˆ i describes interactions between

particles. Additionally, Hˆ 0 is understood to be in the grand canonical ensemble, and therefore we measure energies from the chemical potential µ. The system is initially in thermodynamic

equilibrium with a connected reservoir, and at time t = t0 it is disconnected from the reservoir and subjected to a perturbation represented by the addition of H0(t) to the Hamiltonian. Then the total Hamiltonian is

ˆ = Hˆ + H0(t) , (2.2) H

where H0(t) = 0 for times t < t0. We are interested in calculating physical observables at times

t > t0. In the Heisenberg picture, expectation values of physical observables, ˆ (t), in this system are OH found using the equilibrium density matrixρ ˆ(t0) via

n o (t) = Tr ρˆ(t0) ˆ (t) , (2.3) O OH where t < t, and the entire time-dependence resides in the operator ˆ. The equilibrium density 0 O matrix is given by βHˆ e− ρˆ(t0) = n o , βHˆ Tr e−

1 with β = (kBT)− the usual inverse temperature, and the traces are performed over the many-body CHAPTER 2. FORMALISM 11

Hilbert space. The operator in the Heisenberg picture is given by

ˆ (t) = Uˆ †(t, t0) ˆ(t)Uˆ (t, t0) , (2.4) OH O

and the operator written as ˆ(t) is given in the Schrodinger¨ picture, such that it may possess only O its own explicit time dependence. In the Heisenberg picture, the operator for a physical observable obeys the Heisenberg equation of motion

ι h i ˆ˙ (t) = ˆ (t), Hˆ (t) . OH −~ OH

The evolution operator introduced above is explicitly written

ι R t ˆ ~ dτ (τ) Uˆ (t, t0) = e− t0 H , (2.5) T

and is the time-ordering operator which places all operators with later times to the left of those T at earlier times. The evolution operator is unitary, such that

Uˆ †(t, t0)Uˆ (t, t0) = Uˆ (t0, t)Uˆ (t, t0) = 1 .

We now use Eq. (3.5) along with the time-evolved operator to represent the physical observable as

n o (t) = Tr ρˆ(t )Uˆ †(t, t ) ˆ(t)Uˆ (t, t ) O 0 0 O 0 n o = Tr ρˆ(t )Uˆ (t , t) ˆ(t)Uˆ (t, t ) . (2.6) 0 0 O 0

Viewing from left to right, we see that the system begins in equilibrium at time t0, where the equilibrium density matrix is speciﬁed, and evolves forward to time t at which point the observable is CHAPTER 2. FORMALISM 12

calculated. From there, the system evolves back from time t to the initial equilibrium configuration through the second evolution operator Uˆ . This is our first contour, depicted in Fig. 2.1a, the introduction of which automatically obviates the need to assume anything about the state of the system in the infinite future. However, making use of the unitarity of the evolution operator, we may extend the contour such that

1 (t) = Tr ρˆ(t )Uˆ (t , t) ˆ(t)Uˆ (t, )Uˆ ( , t)Uˆ (t, t ) O 2 0 0 O ∞ ∞ 0 + ρˆ(t )Uˆ (t , t)Uˆ (t, )Uˆ ( , t) ˆ(t)Uˆ (t, t ) , 0 0 ∞ ∞ O 0 R 1 n ι dt ˆ (t ) h + io = Tr ρˆ(t ) e− ~ c 0H 0 ˆ(t ) + ˆ(t−) (2.7) 2 0 Tc O O

where we have now switched to the contour that runs from t = t0 to inﬁnity and back again, denoted by the subscript c. The time ordering operator orders times in the contour sense, such Tc that “later” times are those times further along the contour. The physical observable is now given

+ by the average of the operator evaluated at t on the forward branch (denoted +) and t− on the backward branch (denoted ). That is, we can see here that evolution begins at time t , progresses − 0 to t = , and then back to the initial equilibrium state located at t , which we ostensibly know. ∞ 0 This contour is depicted in Fig. 2.1b.

2.1.1 Adiabatic switching

In order to introduce the Keldysh contour, as depicted in Fig. 2.1c, we will discuss the idea of adiabatic switching. The central concept behind the adiabatic assumption is that it is possible to generate the density matrixρ ˆ of Hamiltonian Hˆ starting from the density matrixρ ˆ0 corresponding to the noninteracting Hamiltonian Hˆ 0 and then “switching on” interactions adiabatically. That is to say, we have ρˆ(t ) = Uˆ (t , )ˆρ Uˆ ( , t ) , (2.8) 0 η 0 −∞ 0 η −∞ 0 CHAPTER 2. FORMALISM 13

(a) t0 t

(b) t0 t+ c

t 1

(c) + t cK 1 t 1 Figure 2.1: Diagram (a) depicts the contour presented by Eq. (2.6), where evolution begins in equilibrium at time t0, progresses forward until time t where the operator is evaluated, and then back to equilibrium. Figure (b) is a cartoon of the contour, denoted c, after extension to t = ∞ by insertion of the unitary evolution operators. Image (c) depicts the Keldysh contour, cK, where evolution proceeds from the inﬁnite past to the inﬁnite future and back, attained after invoking the adiabatic assumption.

where Uˆ η is the real-time evolution operator deﬁned with the Hamiltonian

η t t0 Hˆ η = Hˆ 0 + e− | − |Hˆ i , (2.9) and η is an inﬁnitesimal positive constant. This Hamiltonian becomes the noninteracting Hamilto- nian Hˆ when t , and interactions have been introduced adiabatically such that Hˆ Hˆ by 0 → −∞ η → time t = t0. We may then re-express Eq. (2.6) in terms of the non-interacting density matrix using

Uˆ η as

n o (t) = Tr ρˆ(t )Uˆ (t , t) ˆ(t)Uˆ (t, t ) O 0 0 O 0 n o = Tr ρˆ Uˆ ( , t )Uˆ (t , t) ˆ(t)Uˆ (t, t )Uˆ (t , ) , (2.10) 0 η −∞ 0 0 O 0 η 0 −∞ where note that we have used the cyclic property of the trace in the second line. At this point, we cast Eq. (2.10) as a contour ordered series of operators. Consider the contour depicted in CHAPTER 2. FORMALISM 14

Fig. 2.1c, where time evolution begins in the infinite past, progresses to the infinite future, and then proceeds back to the infinite past once more. Defining the appropriate Hamiltonian that governs the dynamics along this contour, we write

   ˆ η t t0 ˆ H0 + e− | − |Hi t < t0 t =  . H ( )  (2.11)  ˆ ˆ ˆ H0 + Hi + H0(t) t > t0

Thus, via the adiabatic assumption and subsequent adiabatic switching, we ﬁnd the adiabatic formula: ι R h i 1 ~ dt0H (t0) + (t) = Tr T ρˆ e− cK ˆ(t ) + ˆ(t−) , (2.12) O 2 K 0 O O

where TK is the Keldysh time-ordering operator that orders operators in the contour sense, according to their position around the Keldysh time-loop c . The superscripts, + and , denote forward K −

and backward branches of cK respectively. This result holds whenever the adiabatic assumption is fulﬁlled.

2.1.2 The equilibrium limit

A special case is what we can term the equilibrium limit, where there are no driving ﬁelds such

that Hˆ 0(t) = 0, meaning the Hamiltonian becomes independent of time. For any ﬁnite observation

time t we can therefore approximate Uˆ (t, t0) with Uˆ η(t, t0) because it is always possible to select

1 η t t − . As a result, we can write Eq. (2.10) as | − 0|

n o (t) = Tr ρˆ Uˆ ( , t) ˆ(t)Uˆ (t, ) . (2.13) O 0 η −∞ O η −∞

In this case, the Hamiltonian has become

η t t ˆ (t) = Hˆ + e− | − 0|Hˆ Hˆ + Hˆ (t) , (2.14) Hη 0 i ≡ 0 η CHAPTER 2. FORMALISM 15

and is now valid along the entire contour, cK. Thus, we may write the observable using the Keldysh

contour and Hˆ η as ι R ˆ h i 1 ~ dt0 η(t0) + (t) = Tr T ρˆ e− cK H ˆ(t ) + ˆ(t−) . (2.15) O 2 K 0 O O

The standard approach to diagrammatic perturbation theory begins by shifting pictures from the Heisenberg picture to the interaction picture. In the interaction picture operators evolve with the non-interacting Hamiltonian, Hˆ 0, such that

ιHˆ0(t tr) ιHˆ0(t tr) ˆ ˆ (t) = e − ˆ(t)e− − , OH0 O

where tr is a reference time much further in the past than t0, the time at which interactions are fully engaged. Then, we may write expectation values utilizing the interaction picture as

R ι ˆ h i 1 ~ dt0Hη,Hˆ (t0) + (t) = Tr T ρˆ e− cK 0 ˆ ˆ (t ) + ˆ ˆ (t−) , (2.16) O 2 K 0 OH0 OH0

where

ιHˆ (t t ) ιHˆ (t t ) ˆ = 0 − r ˆ − 0 − r Hη,Hˆ0 (t) e Hη(t)e

is the interaction Hamiltonian time-evolved with the non-interacting Hamiltonian. Eq. (2.16) is the main result of this section, and will be used throughout this thesis when calculating expectation values of physical operators, e.g., the spin or charge current in a system.

2.2 Keldysh Green functions

Green functions constitute the connection between experimentally relevant and conveniently cal- culable quantities, and thus play an important role in statistical physics. In this section, we will

introduce the Green functions associated with the contour cK. This will begin by examining the generic time-ordered Green function, introducing the S -matrix, and using the S -matrix to place CHAPTER 2. FORMALISM 16 the Green function on the Keldysh contour. Consider a time-ordered Green function for a system in its full interacting ground state : |i

D h iE G(x, x0) = ι ψ ˆ (x)ψ† (x0) , (2.17) − T H Hˆ

with ψHˆ (x) the field operator in the Heisenberg picture with respect to the full interacting Hamil- tonian Hˆ , and x = (x, t). In the standard approach to evaluating this Green function in equilibrium, zero-temperature quantum field theory we write G(x, x0) in the interaction picture. That is accomplished by writing the field operators, ψ, in the interaction picture via

ιHˆ (t t ) ιHˆ (t t ) ψ = 0 − r ψ x − 0 − r . Hˆ0 (x) e ( )e (2.18)

This time-evolved operator connects back to its Heisenberg picture counterpart through the unitary transform known as the S -matrix:

ψ = † , ψ , , Hˆ (x) S (t tr) Hˆ0 (x)S (t tr) (2.19) and the S -matrix is given by R ι t ˆ ~ dt00Hi,Hˆ (t00) S (t, t0) = e− t0 0 . (2.20) T

Then, we can use this relationship to write G(x, x0) in terms of ﬁeld operators in the interaction picture, which is given to be

, 0 = ι † , ψ ψ† 0 , . G(x x ) S (tm tr) Hˆ0 (x) ˆ (x )S (tm tr) (2.21) − T H0

where tm is whichever time, t or t0, is later. We now bring in the adiabatic assumption. In the inﬁnite past, therefore, the system assumes its non-interacting ground state . We send the reference time to the inﬁnite past, t , and |i0 r → −∞ CHAPTER 2. FORMALISM 17

assume that the full interacting state is achieved at time t = 0 by adiabatically evolving using |i0 the S -matrix: = S (0, ) |i −∞ |i0

The time-ordered Green function can now be expressed in terms of the non-interacting ground state such that

, 0 = ι , , ψ , 0 ψ† 0 0, , . G(x x ) S ( 0) S (0 t) Hˆ0 (x)S (t t ) ˆ (x )S (t 0)S (0 ) (2.22) − −∞ T H0 −∞ 0

Our next assumption is that the the reverse process is also true, namely that when the interactions are adiabatically “switched oﬀ” from time t = 0 the same non-interacting ground state that occurs at time t = is also recovered up to the addition of a phase factor. That is, we assume −∞

ιφ 0 S ( , )S ( , 0) S ( , 0) = e− S ( , )S ( , 0) = h| ∞ −∞ −∞ . h|0 −∞ h|0 ∞ −∞ −∞ S ( , ) h ∞ −∞ i0

Emplacing this result into the time-ordered Green function returns the standard result

ψ ψ† , Hˆ0 (x) ˆ (x0)S ( ) T H0 ∞ −∞ 0 G(x, x0) = ι . (2.23) − S ( , ) h ∞ −∞ i0

One of the central assumptions in the equilibrium, zero temperature quantum field theory approach to G(x, x0) breaks down out of equilibrium in an obvious way. That is, it is no longer reasonable to assume that the non-interacting ground state recovered at time t = is identical to ∞ the original non-interacting ground state the system began in at time t = after the insertion of −∞ a non-equilibrium perturbation. The system may be in an unpredictable state due, for example, to some irreversible effect. The core idea of avoiding this problem by utilizing a contour starting and ending in the infinite past was originally proposed by Schwinger, and subsequently the contour was popularized by Keldysh — thus it achieves its name. Even in equilibrium, however, the time-loop CHAPTER 2. FORMALISM 18

(++) (+ ) t 0 t cK t cK 1 1 1 t0 1

( +) ( ) t0 cK cK 1 1 t 1 t t0 1 Figure 2.2: The different time-ordering configurations possible on the Keldysh contour. Red dots represent field operators at the depicted times, t or t0, on a branch of the contour. approach has the advantage of avoiding the assumption of a known state in the infinite future. The integration path can be seen in Fig. 2.1c, starting from the infinite past, through to the infinite future, and then back along the second leg of the contour to the infinite past once more. We can then define the S matrix on the Keldysh contour, S K, as

R ι ˆ dt0H , ˆ (t0) S ( , ) = T e− ~ cK i H0 , (2.24) K −∞ −∞ K

and use S K to deﬁne the contour-ordered Green function:

, 0 = ι , ψ ψ† 0 . GcK (x x ) TKS K( ) Hˆ0 (x) ˆ (x ) (2.25) − −∞ −∞ H0 0

2.2.1 The Kadanoﬀ-Baym basis

Consider a set of creation and annihilation operators,a ˆα† (t) anda ˆα(t) respectively, where we have depicted the times exclusively because we will be considering time-time correlations in what follows. They are labeled by some complete set of quantum numbers α (which maybe be spin pro- jections, momentum, etc.) and are either bosons or fermions. The contour-ordered Green function presented above admits of four possibilities that must be accounted for: the two situations arising when t and t0 reside on the same branch of the contour and the two possible ways in which the two CHAPTER 2. FORMALISM 19

times can reside on opposite branches of the contour — see Fig. 2.2. The contour ordered Green

function, GcK , can be mapped onto the Keldysh space via

   ++ +  Gˆ Gˆ − ˆ   GcK G =   7→  +  Gˆ− Gˆ−− where the superscripts indicate branches of the contour, + for the outgoing branch from to , −∞ ∞ and for the return branch. The four components are −

++ D + + E D E Gˆ (t, t0) = ι TKaˆα(t )â†(t0 ) = ι aˆα(t)â†(t0) (2.26) αβ − β 0 − T β 0 + D + E D E Gˆ −(t, t0) = ι TKaˆα(t )â†(t0−) = ι aˆ† (t0)âβ(t) (2.27) αβ − β 0 ± α 0 + D + E D E Gˆ− (t, t0) = ι TKaˆα(t−)â†(t0 ) = ι aˆα(t)â†(t0) (2.28) αβ − β 0 − β 0 D E D E Gˆ−−(t, t0) = ι TKaˆα(t−)â†(t0−) = ι ¯ aˆα(t)â†(t0) , (2.29) αβ − β 0 − T β 0

where superscripts in the interim equalities indicate branches of the Keldysh contour, ¯ represents T anti-time-ordering, and the on the + component indicates the result is + ifa ˆ is a fermionic field ± − and if it is a bosonic field. These are the Green functions in what is called the Kadanoff-Baym − basis.

2.2.2 The Keldysh basis

It is clear however that the Green functions in the Kadanoﬀ-Baym basis are not linearly independent. In particular, we can write

++ + + G (t, t0) = θ(t t0)G− (t, t0) + θ(t0 t)G −(t, t0) αβ − αβ − αβ + + G−−(t, t0) = θ(t t0)G −(t, t0) + θ(t0 t)G− (t, t0) , αβ − αβ − αβ CHAPTER 2. FORMALISM 20

where θ(t) is the Heaviside step function. This results in a constraint on the Green functions, such that

++ + + Gαβ (t, t0) + Gαβ−−(t, t0) = Gαβ−(t, t0) + Gαβ− (t, t0) , which we may make use of in order to write a linearly independent set of Green functions. Those Green functions are the retarded, advanced, and Keldysh Green functions given by

R 1 ++ + + + + G (t, t0) = G G−− + G− G − = θ(t t0) G− (t, t0) G −(t, t0) (2.30) αβ 2 αβ − αβ αβ − αβ − αβ − αβ A 1 ++ + + + + G (t, t0) = G G−− G− + G − = θ(t0 t) G −(t, t0) G− (t, t0) (2.31) αβ 2 αβ − αβ − αβ αβ − αβ − αβ K 1 ++ + + + + G (t, t0) = G + G−− + G− + G − = G− (t, t0) + G −(t, t0) , (2.32) αβ 2 αβ αβ αβ αβ αβ αβ

respectively. This set of Green functions are written in the RAK or Keldysh basis, and hold true in both the cases of Bose or Fermi ﬁelds, in and out of equilibrium. These three Green functions serve as the fundamental objects of the Keldysh formalism and will be used extensively in the remainder of this thesis. In the non-interacting limit we can diagonalize the times by transforming to frequency space and the Green functions in the RAK basis become

δαβ h i G0,R(ω) = = G0,A(ω) † (2.33) αβ ω ξ + ι0+ αβ − α G0,K(ω) = 2πι (1 2n (ω)) δ(ω ξ )δ (2.34) αβ − − F − α αβ

in the case of fermions. For bosons, we exchange twice the Fermi-Dirac distribution 2n (ω) for − F + twice the Bose-Einstein distribution, +2nB(ω). The term 0 is meant to indicate an inﬁnitesimal,

positive quantity, and ξα represents the dispersion. The Green functions make clear that the retarded and advanced components carry information about the quantum states and energy spectrum, while the Keldysh component describes the occupation of those states. CHAPTER 2. FORMALISM 21

2.2.3 The ﬂuctuation dissipation theorem

This formulation also contains a statement of the ﬂuctuation dissipation theorem. Speciﬁcally, in thermal equilibrium the Keldysh, retarded, and advanced Green functions maintain a strict relation, namely   tanh ω GR (ω) GA (ω) (for fermions)  2kBT αβ αβ GK (ω) =  − (2.35) αβ   ω R A coth Gαβ(ω) Gαβ(ω) (for bosons) . 2kBT − These dependencies imply a relationship between response and correlation functions that will be utilized in what follows. While it is clear in this non-interacting depiction, the relationship should hold true even when interactions are taken into account in a thermally equilibrated system. Chapter 3

Spin shot noise across coupled 1d spin chains

In this chapter, we apply the Luttinger liquid formulation to a system of weakly exchange coupled 1d quantum paramagnets and utilize the Keldysh path integral formalism to compute the bulk spin-current and spin-current noise downstream of the coupling. The system is driven out-of- equilibrium by both the presence of spin injection at one end and a temperature differential between the coupled paramagnets. We compute the effects of this weak coupling on the bulk transport properties of 1d paramagnets hosting exotic spin excitations as their ground-states, and compare the outcomes to more conventional magnon-hosting, similarly configured 3d insulating magnets. We find a stark difference between the low dimensional and 3d spin systems, where particle-hole symmetry in the low-dimensional system causes temperature driven spin currents across the exchange coupling to vanish. Additionally, we develop an experimental method by which to realize our results, and define the spin Fano factor as a quantity that differentiates between a system comprised of low dimensional quantum paramagnets and that comprised of 3d magnetic insulators. We then compute a physical estimate of the signal strength of these effects and conclude that they should be measurable with currently available devices.

22 CHAPTER 3. 1D SPIN CHAINS 23 3.1 Introduction

The quantification of spin-dependent charge current noise[52; 53; 54; 55; 56; 57; 58; 59; 60] as well as pure spin current noise[61; 62; 63] in mesoscopic conductors has garnered much attention over the past two decades, demonstrating the importance of spin effects on charge transport. In contrast, the study of pure spin current noise in insulating spin systems (i.e., quantum magnets) has received only limited attention. [64; 65; 66] This focal imbalance, however, may soon resolve with the recent pioneering developments in spintronics, where experimentalists are now capable of generating and detecting pure spin currents in insulating magnets using purely electrical signals. [17; 18; 67; 68] In these experiments, two strongly spin-orbit coupled metals are affixed to two opposite ends of a magnetic insulator [e.g., yttrium iron garnet (YIG)], charge current is passed through one metal generating spin current in the magnet via the spin Hall effect (SHE), and charge current is detected in the second metal generated by the inverse SHE. These advancements open doors to the fascinating possibility to quantify spin propagation through quantum magnets via electrical measurements, and render the theoretical investigation of pure spin current noise in these systems timely. A natural setup to study spin current and noise in quantum magnets involves two quantum magnets weakly coupled via the exchange interaction (see, e.g., Fig. 3.1). In the presence of a bias, the exchange coupling allows spin-1 excitations to stochastically tunnel from one system to the other, generating a noisy spin current in the latter. In this context, the physics of spin injection into a quantum magnet should depend on the spin quantum number s of the localized spins. If a spin-1 excitation is injected into an s = 1/2 quantum magnet, a second spin-1 excitation cannot be injected at the same site, generating a partial blockade (or Pauli blockade) during spin injection associated with the fermionic nature of the spin-1/2 operators. [69] Pauli blockade should be absent in large-s quantum magnets, where an approximate theoretical description of the injection process in terms of tunneling bosonic quasiparticles (i.e., magnons) is appropriate. This crossover from CHAPTER 3. 1D SPIN CHAINS 24 z I,S spin current spin accumulation x = x1 y x T1 ⊗ T2 Jc ⊗ injector source chain drain chain drain metal x metal 0 x =0 charge current

Figure 3.1: Two semi-infinite quantum spin chains, one labeled the source and the other the drain, are coupled at their finite ends. Strongly spin-orbit coupled metals act as injector and drain for spin current. The source chain is held at temperature T1 and the drain chain at T2. The spin injection from the injector metal via the spin Hall effect leads to a non-equilibrium accumulation of spin in the source chain. Depicted positions are continuum variables. boson-like to fermion-like spin injection physics as s approaches the quantum limit should have an effect on the tunneling spin current and noise and have direct experimental consequences on spin transport. In this chapter, we compare spin transport across weakly-coupled s = 1/2 quantum magnets to that across weakly-coupled large-s magnetic insulators and evaluate quantities that differentiate between the two: the spin current, its dc noise, and the spin Fano factor defined as the noise-to-signal ratio of the spin current. Specifically, for a quantum magnet we consider the s = 1/2 antiferromagnetic quantum spin chain (QSC) due to its amenability to rigorous theoretical analysis [70; 71] and relevance to real materials. [72; 73; 74; 75; 76; 12; 19] We consider spin currents generated by an over-population of spin excitations in one quantum magnet (i.e., subsystem) relative to the other while simultaneously applying a temperature difference between the two subsystems. We find a strikingly different behavior in the spin Fano factor between the QSC and the large-s (magnon) cases. Unlike the magnon case, a vanishing spin current and concomitant diverging spin Fano factor is found only in the QSC scenario even in the presence of a large temperature bias, provided no over-population of spin excitations in one subsystem relative to other exists. We show that this finding is in exact analogy with the results obtained in the physics of local electron scattering between edge states of two fractional quantum Hall liquids. [77; 78] We also compare the noise generated CHAPTER 3. 1D SPIN CHAINS 25 by non-equilibrium injection of spin into one subsystem to that generated by injecting spins into both subsystems, and find that over-populating both subsystems actually reduces the noise in the QSC scenario but increases the noise in the magnon case. We attribute this suppression in the noise to Pauli blockade physics associated with the fermionic nature of spin-1/2 operators in the QSC scenario. We finally discuss how the spin Fano factor may be experimentally obtained via the inverse SHE used in spintronics. This chapter is organized as follows. Sec. 3.2 introduces our model for the quantum spin chain, wherein we describe the Luttinger model and further reveal a striking similarity between our problem and a similar situation in fractional quantum Hall physics. In Sec. 3.3, we utilize a general non-equilibrium scheme based on the Keldysh formalism to derive spin transport properties in our system of interest, and contrast it to the same calculation in a geometrically equivalent large-s magnonic system. We discuss our results in Sec. 3.4, and present a possible method for experimental verification in Sec. 3.5. Finally, we conclude in Sec. 3.6 and propose potential avenues for future exploration.

3.2 Model

Our system of interest is comprised of two s = 1/2 antiferromagnet spin chains exchange-coupled end-to-end, one labeled the source chain (ν = 1) and the other the drain chain (ν = 2), with each additionally coupled to a metal with strong spin-orbit coupling (e.g., Pt, Ta, etc.) at the other ends (see Fig. 3.1). To allow for possible spin transfer across the spin chains, we consider elevating the temperature of one QSC relative to the other (i.e., thermal bias) and/or injecting a z polarized spin current (hereafter simply referred to as spin current) into the source QSC (i.e., chemical bias). The spin current injection can be facilitated, e.g., by SHE at the upstream end of the chain, where the injector metal is coupled and driven by a charge current. Due to spin-orbit coupling in the metal, a charge current flowing in the y direction can give rise to a spin current flowing in the x CHAPTER 3. 1D SPIN CHAINS 26 direction with spin polarized in the z direction and produce a (z-polarized) spin accumulation at the interface (as shown in Fig. 3.1). [79; 23] Interfacial exchange interaction then allows spin angular momentum to be transferred from the metal’s electron spins to the spin moments in the QSC, effectively leading to an injection of spin current into the QSC. [17; 18; 67; 68; 80; 81; 82] The injected spin current tunnels across the QSCs and is eventually ejected into the drain metal, where the spin current converts into a transverse charge current via the inverse SHE and can therefore be detected electrically. Our focus will be on the spin current flowing at x = x1 in the drain chain and its dc noise (see Fig. 3.1), which should be electrically detectable using the drain metal.

3.2.1 Luttinger model for the spin chains

We consider two identical s = 1/2 semi-inﬁnite xxz antiferromagnet chains that are weakly exchange-coupled at their ﬁnite ends, i.e., at site j = 0 (or x = 0), as shown in Fig. 3.1. The P total Hamiltonian for the QSCs can then be written as Hˆ = Hˆ + Hˆ + Hˆ Hˆ + Hˆ + Hˆ , ν=1,2 ν b c ≡ 0 b c where the Hamiltonians of the two QSCs read

X n o ˆ ˆ x ˆ x ˆ y ˆ y ˆ z ˆ z Hν = J S ν, jS ν, j+1 + S ν, jS ν, j+1 + ∆S ν, jS ν, j+1 , (3.1) j with J, ∆ > 0, and the coupling Hamiltonian reads

n o ˆ ˆ x ˆ x ˆ y ˆ y z ˆ z ˆ z Hc = Jc⊥ S 1,0S 2,0 + S 1,0S 2,0 + JcS 1,0S 2,0 , (3.2)

z where we assume J⊥ , J J and we are allowing for an xxz anisotropy in the exchange cou- | c | | c| pling. The bias Hamiltonian Hˆ b will be speciﬁed later. We assume ∆ < 1 such that the QSCs are in the xy phase with unaxial symmetry and possess a gapless excitation spectrum. [71] In this regime and in the long-wavelength limit, Eq. (3.1) is CHAPTER 3. 1D SPIN CHAINS 27

well-described by a Luttinger liquid Hamiltonian [83; 84; 11]

Z 0 Z ~u n 2 2o ~u ∞ n 2 2o Hˆ 0 = dx [∂xϕ˜ˆ 1,R(x)] + [∂xϕ˜ˆ 1,L(x)] + dx [∂xϕ˜ˆ 2,R(x)] + [∂xϕ˜ˆ 2,L(x)] , (3.3) 4πK 4πK 0 −∞ where u is the speed of the chiral boson fields, K is the so-called Luttinger parameter, and the chiral boson fields obey [ϕ˜ˆ (x), ϕ˜ˆ (x0)] = [ϕ˜ˆ (x), ϕ˜ˆ (x0)] = iπKδ sgn(x x0). [83] Arriving at ν,R ν0,R − ν,L ν0,L νν0 − Eq. (3.3) requires that we drop RG-irrelevant operators (e.g., band-curvature and backscattering terms) and that we constrain our inquiry to the Gaussian regime. In order to remain within the Gaussian regime we find u and K perturbatively by taking ∆ 1. The result is that u = v /K F and K 1 2∆/π for the low energy sector. However, a Bethe ansatz approach shows that the ' − Gaussian model in fact holds for the entire critical domain ∆ < 1 provided we identify u = | | 2 1 1 1 πv √1 ∆ /2 cos− (∆) and K = [2 (2/π) cos− (∆)]− . [85] Therefore, given this exact solution, F − − we assume throughout that ∆ . 1 and so take u πv /2 and K & 1/2, i.e., close to the Heisenberg ' F limit with ∆ . 1. Spin injection at the upstream end of the source QSC should generate a spin accumulation in the QSC, which, in the long-time (steady-state) limit, can be modeled as a (uniform) spin chemical potential µ that extends over its entire length, causing the spins to precess about the z axis and driving current across the coupling. The Hamiltonian describing the chemical bias may then be written as µ Z 0 Hˆ = dx ∂ (ϕ˜ˆ , + ϕ˜ˆ , ) . (3.4) b 2π x 1 R 1 L −∞

3.2.2 Exact mapping to the problem of stochastic electron tunneling be-

tween two fractional quantum Hall edge channels

In order to treat the semi-inﬁnite chains we must account for the ﬁnite boundary. We begin this

process by first introducing new scaled chiral fieldsϕ ˆ ν,R/L = ϕ˜ˆ ν,R/L/ √K in Eq. (3.3), allowing us to map this system onto an effectively non-interacting (i.e., K = 1 or free fermion) Luttinger liquid CHAPTER 3. 1D SPIN CHAINS 28

governed by the Hamiltonian

Z 0 Z ~u n 2 2o ~u ∞ n 2 2o Hˆ 0 = dx [∂xϕˆ 1,R(x)] + [∂xϕˆ 1,L(x)] + dx [∂xϕˆ 2,R(x)] + [∂xϕˆ 2,L(x)] , (3.5) 4π 4π 0 −∞

where now the scaled chiral ﬁelds obey [ϕ ˆ (x), ϕˆ (x0)] = [ϕ ˆ (x), ϕˆ (x0)] = iπδ sgn(x x0). ν,R ν0,R − ν,L ν0,L νν0 − Then the semi-inﬁnite boundary conditions at x = 0 requires thatϕ ˆ (0) = ϕˆ (0), which further ν,R − ν,L enforces that the string operator cos ( ) 1 at the end. 1 We can extend the x = 0 result to ··· → include all space and time by noting that right-movers are a function of x ut only and left-movers − are a function of x + ut only. Thus we have

ϕˆ ( x, t) = ϕˆ (x, t) . (3.6) ν,R − − ν,L

We now impose Eq. (3.6) explicitly on Eq. (3.5) and reinstate the unscaled fields ϕ˜ˆ ν,R(x) such that X Z ~u ∞ 2 Hˆ = dx [∂ ϕ˜ˆ ν, (x)] . (3.7) 0 4πK x R ν=1,2 −∞ We note that the remaining right chiral fields now reside on an infinite domain and obeys the

commutation relation [ϕ˜ˆ (x), ϕ˜ˆ (x0)] = iπKδ sgn(x x0); they can be expanded in terms of ν,R ν0,R νν0 − canonical boson operators as

r X ηk/2 2πK e− ikx ikxo ϕ˜ˆ (x) = i bˆ e bˆ† e− , (3.8) ν,R − L ν,k − ν,k k>0 √k where η is a UV cutoﬀ and L is the chain length (eventually taken to inﬁnity). The boson operator ˆ ˆ P ˆ ˆ bν,k diagonalizes Eq. (3.5) as Hν = k>0 εkbν,† kbν,k with εk = ~uk.

1For a full explication of a ﬁnite edge in a Luttinger liquid, see section 10.1 of Ref. [11]. CHAPTER 3. 1D SPIN CHAINS 29

Explicitly implementing Eq. (3.6) on the spin chemical bias term Eq. (3.4), we obtain

"Z 0 Z # µ ∞ Hˆ b = dx ∂xϕ˜ˆ 1,R(x) + dx ∂xϕ˜ˆ 1,R(x) . (3.9) 2π 0 −∞

Finally, applying Eq. (3.6) on the bosonized spin operators we ﬁnd

√a γν,R + γν,L i ϕˆ + ˆ K ˜ν,R(0) ˆ S ν,−0 = p e = (S ν,0)† , (3.10) 2πη

where a is the lattice constant for the spin chain, γν are Majorana ﬁelds that obey the anti- commutation relation γ , γ = 2δ . Using Eq. (3.10), the coupling Hamiltonian Eq. (3.2) can { µ ν} µν now be re-expressed as

i [ϕ˜ˆ1,R(0) ϕ˜ˆ2,R(0)] Hˆ c = ξ e K − + h.c., (3.11) ⊥ where ξ (Jc⊥a/4πη)(γ1,R+γ1,L)(γ2,R+γ2,L). Equation (3.11) should in principle contain the z com- ⊥ ≡ z ponent of the exchange coupling that gives rise to a term proportional to Jc[∂xϕ˜ˆ 1,R(0)][∂xϕ˜ˆ 2,R(0)].

However, a leading-order RG analysis gives that the scaling dimension for the coupling ξ⊥ is 1 1/K while that for Jz is 1. Since we assume K > 1/2, the latter term is less RG-relevant − c − than the terms appearing in Eq. (3.11) so in the long-wavelength low-energy limit, the inter-chain coupling should be dominated by the transverse components of the exchange coupling presented in Eq. (3.11). We note that Eqs. (3.7), (3.9) and (3.11) exactly correspond to a theoretical model describing two-terminal charge transport between the edge channels of two “Laughlin” fractional quantum Hall liquids at ﬁlling fraction K (see Fig. 3.2). [78] In the fractional quantum Hall scenario, Eq. (3.11) describes a stochastic tunneling of (charge e) electrons (with amplitude ξ ) between − ⊥ the two edge states, and the role of the spin chemical bias µ is played by the (electrical) voltage bias µ = eV applied between the two edge channels. Equation (3.7) resembles the edge state Hamil- tonian for the Laughlin fractional quantum Hall liquids, but we note that for the actual Laughlin CHAPTER 3. 1D SPIN CHAINS 30

fractional quantum electron tunneling fractional quantum µ Hall liquid at filling Hall liquid at filling fraction K ⇠ fraction K

Figure 3.2: The problem of tunneling spin-1 excitations between two semi-inﬁnite QSCs exactly maps to the problem of stochastic electron tunneling between the edge states of two Laughlin fractional quantum Hall liquids at ﬁlling fraction K. The role of the spin chemical bias in the QSC problem is played by the electrical voltage µ = eV applied to the edge channel on the left in the quantum Hall case. states, K is directly determined by the topological property of the bulk quantum Hall state and is constrained to inverse odd integers. [78]

3.3 Spin current and dc noise

We now focus on the spin current ﬂowing in the drain chain and its dc noise at the spatial point

x = x1 just to the left of the drain metal (see Fig. 3.1). These transport quantities may be measured electrically in the drain metal. The spin current can be obtained by evaluating the Keldysh expectation value of the operator for spin current [after implementing the boundary condition (3.6)]

~ X X ˆ u ˆ I(x1, t) = lim ρ∂xϕ˜ˆ 2,R(ρx, t) Iρ(x1, t) . (3.12) x x1 π → 2 ρ= ≡ ρ= ± ±

In the inﬁnite past, we assume the source (drain) QSC to be in a thermal state with respect to Hˆ 1+Hˆ b

(Hˆ 2) at temperature T1 (T2) and that the two chains are isolated. The coupling between the QSCs

Hˆ c is then introduced adiabatically and treated perturbatively within the Keldysh diagrammatic CHAPTER 3. 1D SPIN CHAINS 31

approach. [86; 87] The current expectation value on the Keldysh contour reads

1 X X σ 1 X X I(x1, t) = Iˆρ(x1, t ) Iρσ , (3.13) 2 ρ= σ= h i ≡ 2 ρ= σ= ± ± ± ±

where σ = labels the time on the forward (+) and return ( ) contour. The dc noise is given by ± − the time-averaged autocorrelation function of current at diﬀerent times

X Z ˆ ˆ + S (x1) = dt Iρ1 (x1, t−)Iρ2 (x1, 0 ) . (3.14) ρ , ρ = h i 1 2 ±

3.3.1 Spin current

We begin by computing the spin current ﬂowing at x = x1 which can be reconstructed from its components according to Eq. (3.13),

i R σ dtHˆc(t) I = T Iˆ (x , t )e− ~ cK , (3.15) ρσ h K ρ 1 i0

where cK indicates a time integral on the Keldysh contour and TK is the Keldysh time order-

2 2 ing operator. By expanding to order ξ [i.e., (Jc⊥) ], using Eq. (6.4) and noting that it is possi- ⊥ ble to write all quantities in terms of exponentiated boson operators via the relation ∂xϕˆ(x, t) =

1 iγϕˆ(x,t) limγ 0(iγ)− ∂xe , we express Eq. (3.15) as → Z Z ρ σ iu 1 iγϕ˜ˆ2,R(ρx,t ) Iρσ = lim lim dt1 dt2∂x TKe Hˆ c(t1)Hˆ c(t2) 0 . (3.16) π~ x x1 γ 0 γ 4 → → cK cK h i

Expanding the coupling Hamiltonians using Eq. (3.11), we obtain

2 iuρξ σ i 1 iµ(t1 t2)/~ iγϕ˜ˆ2,R(ρx,t ) [ϕ˜ˆ2,R(0,t1) ϕ˜ˆ2,R(0,t2)] Iρσ = ⊥ lim lim dt1dt2 e− − ∂x TKe − K − 0 2π~ x x1 γ 0 γ h i → → cK (3.17) i " [ϕ˜ˆ , (0,t ) ϕ˜ˆ , (0,t )] T e K 1 R 1 − 1 R 2 , × h K i0 CHAPTER 3. 1D SPIN CHAINS 32

and note here that we have used the fact that the spin bias appears as a phase attached to the

correlator for ϕ˜ˆ 1,R. We now introduce the (equilibrium) time ordered and anti-time ordered correlation functions

i ++ [ϕ˜ˆν, (0,t ) ϕ˜ˆν, (0,t )] D (t , t ) Te K R 1 − R 2 ν 1 2 ≡ h i0|µ=0 + + = θ(t1 t2)Dν− (t1, t2) + θ(t2 t1)Dν −(t1, t2) , − − (3.18) i [ϕ˜ˆν, (0,t ) ϕ˜ˆν, (0,t )] D−−(t , t ) Te¯ K R 1 − R 2 ν 1 2 ≡ h i0|µ=0 + + = θ(t t )D− (t , t ) + θ(t t )D −(t , t ) , 2 − 1 ν 1 2 1 − 2 ν 1 2

where T (T¯) is the time (anti-time) ordering operator, θ is the Heaviside function and

 1/K  πkBTν η   u~  Dν∓±(t1, t2) =  π  . (3.19) sin kBTν [ iu(t t ) + η] u~ ± 1 − 2

We can then write Eq. (3.17) as

2 Z Z iuρξ X ∞ ∞ γ σ σ1 γ σ σ2 1 TK ϕ˜ˆ2,R(ρx,t )ϕ˜ˆ2,R(0,t ) 0 TK ϕ˜ˆ2,R(ρx,t )ϕ˜ˆ2,R(0,t ) 0 Iρσ = ⊥ lim lim σ1σ2 dt1 dt2∂xe K h 1 i e− K h 2 i π~ x x1 γ 0 γ 2 → → σ ,σ = 1 2 ± −∞ −∞

σ1σ2 σ1σ2 iµ(t t )/~ D (t , t )D (t , t )e− 1− 2 , × 1 1 2 2 1 2 (3.20) and the Keldysh contours determine the directionality of the D functions. Performing the γ 0 ν → limit, we have

2 Z Z iuξ X ∞ ∞ σσ1 σσ2 Iρσ = lim ⊥ σ1σ2 dt1 dt2∂x[ fρx (t, t1) fρx (t, t2)] x x1 π~ → 2 K σ ,σ = − 1 2 ± −∞ −∞ (3.21)

σ1σ2 σ1σ2 iµ(t t )/~ D (t , t )D (t , t )e− 1− 2 , × 1 1 2 2 1 2 where f σ1σ2 (t , t ) = T ϕ˜ˆ (ρx, tσ1 )ϕ˜ˆ (0, tσ2 ) . It is useful at this point to introduce the coordi- ρx 1 2 h K 2,R 1 2,R 2 i0 nates τ = 1 (t + t ) and τ = t t , and shift the resultant integrals in order to cancel out the terms 0 2 1 2 1 − 2 in Eq. (3.21) with σ = σ . We additionally see that the remaining two terms (with σ = σ ) can 1 2 1 − 2 CHAPTER 3. 1D SPIN CHAINS 33

be made identical up to their phase factors by interchanging the variables t t for one of the 1 ↔ 2 + + terms and noting that D −( τ) = D− (τ). We ﬁnd that Eq. (3.21) then reduces to ν − ν

2 Z Z uξ ∞ µτ + + ∞ h σ σ+ i Iρσ = lim ⊥ dτ sin D1− (τ)D2− (τ) dτ0 ∂x fρx−(0, τ0) fρx (0, τ0) , (3.22) − x x1 π~K ~ − → −∞ −∞ where the two integrals have been decoupled by the coordinate transformation. Lastly, we complete the τ0 integral by noting

h + ++ i h + i 2πiK ∂ f −(0, τ ) f (0, τ ) = ∂ f −−(0, τ ) f − (0, τ ) = θ( τ )δ(ρx/u + τ ) , (3.23) x ρx 0 − ρx 0 x ρx 0 − ρx 0 − u − 0 0 which allows us to produce the ﬁnal result for this calculation

2 Z 2iξ ∞ µτ + + I = ⊥ θ(ρx ) dτ sin D− (τ)D− (τ) . (3.24) ρσ ~ 1 ~ 1 2 −∞

From Eq. (3.13), we then write the ﬁnal result for the bulk spin current at an arbitrary point x1 > 0 in the drain chain as 2 Z 2iξ ∞ µτ + + I(x , t) = ⊥ dτ sin D− (τ)D− (τ) . (3.25) 1 ~ ~ 1 2 −∞

The Heaviside function appearing in the expression for Iρσ is a manifestation of causality: we expect no contribution from the ρ = portion in our case, as there is no coupling interaction, i.e. − current tunneling into the drain chain, until the point x = 0.

3.3.2 Dc noise

The dc noise calculation proceeds similarly. We start from the general expression for the noise at

x = x1, Z X i R X + dtHc(t) ˆ ˆ − ~ cK S (x1) = dt TK Iρ1 (x1, t−)Iρ2 (x1, 0 )e 0 S ρ1ρ2 . (3.26) ρ , ρ = h i ≡ ρ , ρ = 1 2 ± 1 2 ± CHAPTER 3. 1D SPIN CHAINS 34

and expand this expression up to second order in ξ . At zeroth order, using Eqs. (3.8) and (6.4), ⊥ we obtain Z (0) + ~KkBT2 S = dt T Iˆρ (x , t−)Iˆρ (x , 0 ) = . (3.27) ρ1ρ2 h K 1 1 2 1 i0 2π

So the equilibrium (Johnson-Nyquist) contribution to the dc spin current noise at x = x1 is given

(0) by S (x1) = 2~KkBT2/π. The ﬁrst non-trivial correction to this equilibrium result comes at second order in ξ . Rep- ⊥ resenting the current operators in exponentiated form as in the spin current calculation, the non-

(2) P (2) equilibrium correction reads S (x1) = ρ ,ρ = S ρ1ρ2 , where 1 2 ±

2 2 X Z Z Z (2) u ξ ρ1ρ2 ∞ ∞ ∞ S = lim lim ⊥ σ σ dt dt dt ρ1ρ2 2 1 2 1 2 x,y x1 γ1,γ2 0 π γ γ → → 4 1 2 σ ,σ = 1 2 ± −∞ −∞ −∞ (3.28) + i σ1 σ2 i σ1 σ2 iγ ϕ˜ˆ , (ρ x,t )+iγ ϕ˜ˆ , (ρ y,0 ) [ϕ˜ˆ , (0,t ) ϕ˜ˆ , (0,t )] [ϕ˜ˆ , (0,t ) ϕ˜ˆ , (0,t )] ∂ ∂ T e 1 2 R 1 − 2 2 R 2 − K 2 R 1 − 2 R 2 T e K 1 R 1 − 1 R 2 . × x yh K i0h K i0

We employ Eq. (4.18) and f σ1σ2 (t , t ) as deﬁned above and perform the γ , γ 0 limits to obtain ρx 1 2 1 2 →

2 2 Z Z Z ξ ρ ρ ∞ ∞ ∞ X (2) u 1 2 iµ(t t )/~ σ1σ2 σ1σ2 S = lim ⊥ dt dt dt σ σ e− 1− 2 D (t , t )D (t , t ) ρ1ρ2 2 2 1 2 1 2 1 1 2 2 1 2 x,y x1 π → 4 K σ ,σ = −∞ −∞ −∞ 1 2 ± h σ σ ih +σ +σ i ∂ ∂ f − 1 (t, t ) f − 2 (t, t ) f 1 (0, t ) f 2 (0, t ) . × x y ρ1 x 1 − ρ1 x 2 ρ2y 1 − ρ2y 2 (3.29)

Expanding the Keldysh contours and noting again that the contributions from σ1 = σ2 vanish, we have

2 2 Z Z Z " # (2) u ξ ρ1ρ2 ∞ ∞ ∞ µ(t1 t2) + + S = lim ⊥ dt dt dt cos − D− (t , t )D− (t , t ) ρ1ρ2 2 2 1 2 1 1 2 2 1 2 x,y x1 − 2π K ~ → −∞ −∞ −∞ (3.30) h + ih + ++ i ∂ ∂ f −−(t, t ) f − (t, t ) f −(0, t ) f (0, t ) . × x y ρ1 x 1 − ρ1 x 2 ρ2y 1 − ρ2y 2 CHAPTER 3. 1D SPIN CHAINS 35

Once again, we can transform to the coordinates τ0 and τ and ﬁnd

2 2 Z Z (2) u ξ µτ + + h + i S = lim ⊥ dτ cos D− (τ)D− (τ) dt f −−(t, 0) f − (t, 0) ρ1ρ2 2 2 1 2 ρ1 x ρ1 x x,y x1 − π ~ − → 2 K Z (3.31) h + ++ i dτ f −(0, τ ) f (0, τ ) , × 0 ρ2y 0 − ρ2y 0

where we note that the three integrals have decoupled as in the spin current case. Equation (3.23) allows us to proceed and we obtain

Z (2) 2 µτ + + S ρ ρ = 2ξ θ(ρ1 x1)θ(ρ2 x1) dτ cos D1− (τ)D2− (τ) . (3.32) 1 2 ⊥ ~

Once more, as for the spin current, the Heaviside functions are manifestations of causality: we expect no non-equilibrium noise in the system until points after the tunneling site at x = 0. Therefore, our ﬁnal results for the spin current and dc noise at x = x1 are

2 2 Z i(Jc⊥) a µτ + + I(µ, T , T ) = dτ sin D− (τ)D− (τ) (3.33) 1 2 2~π2η2 ~ 1 2 and 2 2 Z 2~KkBT2 (Jc⊥) a µτ + + S (µ, T , T ) = + dτ cos D− (τ)D− (τ) , (3.34) 1 2 π 2π2η2 ~ 1 2

respectively.

While we have presented a Keldysh calculation for the spin current and noise at x = x1, Eqs. (3.33) and (3.34) could have been obtained instead by computing the tunneling spin current and its noise at the coupling site x = 0. We have veriﬁed that this latter calculation results in a current that is identical to Eq. (3.33) and a noise that is identical to the second term in Eq. (3.34). This outcome is physically sensible. As is apparent from Eq. (3.7), the QSCs are modeled as an essentially free boson gas. Therefore, non-equilibrium disturbances produced at the upstream end should remain unmodiﬁed as they propagate downstream to x = x1. In particular, tunneling spin current at the coupling site x = 0 should be identical to the current downstream. Moreover, any ad- CHAPTER 3. 1D SPIN CHAINS 36

spin current spin accumulation Im,Sm

⊗ µ, T1 µ =0,T2 injector Im0 ,Sm0 drain metal source bath drain bath metal charge current Figure 3.3: Depiction of proposed magnonic system. Two coupled magnon baths are held at temperatures T1 and T2 and chemical potentials µ1 and µ2, respectively. ditional noise generated at the left end of the drain QSC should propagate downstream undisturbed. We will be using this fact in the proceeding magnon transport comparison calculation.

3.3.3 Spin transport between two magnon baths

Equations (3.33) and (3.34) can now be contrasted against the case of spin transport across two coupled magnon baths (see Fig. 3.3). In similar spirit to the QSC calculation, our interest here is the spin current Im flowing at the spatial point just left of the drain metal (indicated by the black arrow) and its dc noise S m, as these are the physical quantities that may be electrically detected by the metal. We consider both raising the magnon chemical potential in the source bath above zero and/or the spin Seebeck effect, [79] in which spin current is generated by a temperature difference between the two baths. SHEs can be utilized to transfer spin angular momentum from the injector metal into the source magnon bath and raise the magnon chemical potential in the latter; [18; 67; 88; 89] the injection process is analogous to the spin injection process discussed in the context of the QSC setup (see Sec. 3.2). In the absence of the bath coupling, we assume that magnon bath ν is thermalized to a distri-

1 bution function n (Ω) = exp[β (~Ω µ )] 1 − , where ν = 1, 2 labels the two magnon baths, ν { ν − ν − } 1 βν = (kBTν)− is the inverse temperature, and µν is the magnon chemical potential; we set µ1 = µ and µ2 = 0 throughout unless otherwise stated. CHAPTER 3. 1D SPIN CHAINS 37

We work with two identical spin-s Heisenberg ferromagnetic insulators (with N N N x × y × z cubic lattice structures) in a uniform magnetic ﬁeld along the z axis

J X X Hˆ m = Sˆ Sˆ + H Sˆ z , (3.35) ν − ν, j · ν, j+δ ν, j 2 j,δ j

where J is the exchange coupling, H is the Zeeman energy due to an external magnetic ﬁeld along the z axis, j labels sites of the lattice, and δ labels all nearest neighbor sites (this model has been applied to, e.g., YIG with s 14 and lattice constant a 12Å [90]). Assuming s 1, Eq. (3.35) ≈ ≈ z maps to an essentially non-interacting boson model via Sˆ − √2sbˆ and Sˆ = bˆ† bˆ s. [91] ν, j ≈ ν, j ν, j ν, j ν, j − The system is comprised of two semi-inﬁnite systems coupled along the x = 0 plane, thus we consider the boundary and so diagonalize Eq. (3.35) via the Fourier transform

s X 2 iky jya+ikz jza bˆν, j = e cos(kx jxa)bˆν,k , (3.36) NxNyNz k

which is appropriate for a zero-ﬂux boundary condition at the interface; the momenta are kx = ˆ m P ˆ ˆ πnx/Nxa and ky,z = 2πny,z/Ny,za, with ni = 1,..., Ni. We then arrive at Hν = k εkbν,† kbν,k with magnon dispersion ε = 2Js[cos(k a) + cos(k a) + cos(k a) 3] + H. Finally, we use a general k − x y z − interfacial exchange coupling

X m ˆ † Hc = s Jckpbk,1bp,2 + h.c.. (3.37) k,p

We note that for s 1 as assumed here, Eq. (3.35) maps to an essentially free boson model. Therefore, as discussed at the end of Sec. 3.3.2, tunneling spin current across the two magnon baths and its noise (denoted by Im0 and S m0 , respectively, in Fig. 3.3) enter the drain bath at x = 0 and should propagate undisturbed down to the drain metal, where they can be detected. We therefore

(0) (0) expect here that Im = Im0 and S m = S m + S m0 , where S m is the equilibrium (Johnson-Nyquist) spin current noise present in the drain magnon bath even in the absence of Jc. Here, we present CHAPTER 3. 1D SPIN CHAINS 38

calculations for Im0 and S m0 . The operator for spin current tunneling across the two baths can be deﬁned as the total spin leaving bath 1, X ˆ ˆ z Im0 (t) = ~∂t S 1, j . (3.38) − j

ˆ Computing the non-equilibrium expectation value of Im0 to lowest non-trivial order in Jc, we may collapse the Keldysh contour and represent the result more compactly as

Z i m I0 = dt θ( t) [Iˆ0 (0), Hˆ (t)] , (3.39) m −~ − h m c i0 where depicts an equilibrium average with respect to the above-mentioned thermal states. hi0

Again, to lowest non-trivial order in Jc, the dc noise is given by the time-averaged autocorrelation function of current at diﬀerent times, i.e.,

Z S 0 = dt Iˆ0 (t)Iˆ0 (0) . (3.40) m h m m i0

> Denoting the “greater” and “lesser” Green functions for the magnons via G (t) = i b (t)b† (0) k,ν − h k,ν k,ν i0 < and G (t) = i b† (0)b (t) , we then obtain k,ν − h k,ν k,ν i0

2 2 Z Jc s X n < > > < o I0 = dΩ G (Ω)G (Ω) G (Ω)G (Ω) , (3.41) m − π~ k,1 p,2 − k,1 p,2 2 k,p 2 2 Z Jc s X n < > > < o S 0 = dΩ G (Ω)G (Ω) + G (Ω)G (Ω) , (3.42) m − π k,1 p,2 k,1 p,2 2 k,p

where we have taken J J . The Fourier transformed Green functions are G< (Ω) = 2πin (Ω)δ(Ω c ≡ ckp k,ν − ν − ε /~) and G> (Ω) = 2πi[1 + n (Ω)]δ(Ω ε /~), where n is the Bose distribution deﬁned above k k,ν − ν − k ν and ε Js(ka)2 + H is the magnon dispersion. k ≈ CHAPTER 3. 1D SPIN CHAINS 39

Then the spin current and its noise downstream near the drain metal should read

Z 1 Im(µ, T1, T2) = dΩ g(Ω)[n1(Ω) n2(Ω)] , (3.43) ~ H/~ − Z (0) S m(µ, T1, T2) = S m + dΩ g(Ω)[n1(Ω) + n2(Ω) + 2n1(Ω)n2(Ω)] , (3.44) H/~ where g(Ω) = (J ~N N N )2(~Ω H)/8π3 J3 s encodes the magnon tunneling density of states. The c x y z − (0) exact expression for the equilibrium noise component S m will not be essential in the remainder of the discussion.

3.4 Discussion

3.4.1 Spin Fano factor

We begin by fixing the temperature of subsystem ν = 2 to kBT2/J = 0.004 in the QSC case (e.g., T 10 K for Sr CuO with J 2000 K [19]) and k T /Js = 0.08 in the magnon case (e.g., T 2 ∼ 2 3 ≈ B 2 2 ∼ 4 K for YIG [90]). We define the dimensionless thermal bias τ (T T )/T in both cases and the ≡ 1 − 2 2 non-equilibrium noise S (µ, T , T ) S (µ, T , T ) S [S (µ, T , T ) S (µ, T , T ) S neq 1 2 ≡ 1 2 − eq m,neq 1 2 ≡ m 1 2 − m,eq for the magnon case], where S S (µ = τ = 0) [S S (µ = τ = 0)] is the background eq ≡ m,eq ≡ m (thermal) noise in the absence of any bias. Fig. 3.4 then depicts the spin Fano factor, defined as F S /~I and F S /~I as a function of the chemical bias µ for various temperature ≡ neq m ≡ m,neq m biases τ.

From Eq. (3.43), we see that a finite magnon current Im can be generated with either a finite µ or a finite τ, and we find Fm = 1 for any µ and/or τ. The spin Fano factor defined here corresponds only to the non-equilibrium contribution to the spin current noise. Therefore, Fm = 1 reflects the (uncorrelated) Poissonian tunneling of magnons, each carrying a spin quantum of ~, generated by the non-equilibrium biases. 2

2Ref. [64] has calculated shot noise in the spin current generated via spin pumping with a mono-domain ferromag- CHAPTER 3. 1D SPIN CHAINS 40

magnon results for all temperatures 1 QSC results

⌧data1=0.50 ⌧data2=0.25 Spin Fano Factor ⌧data3=0

0 0.05 0.10 0.15

µ/J

Figure 3.4: Spin Fano factor in the QSC and magnon cases plotted as a function of the chemical bias µ µ . Here, τ = (T T )/T , and we set µ = 0 and T T in both systems. ≡ 1 1 − 0 0 2 2 ≡ 0

The QSC spin Fano factor behaves markedly different. As µ increases, i.e., enters the regime µ k T , the QSC spin Fano factor approaches 1, the same value as the magnon spin Fano B 2 factor. This shows that for large biases (i.e., in the shot limit), spin current across the two QSCs is mediated by a Poissonian tunneling of spin-1 excitations, consistent with the inter-chain exchange coupling Eq. (3.2) which transfers spin-1 excitations across the QSCs. However, the QSC spin Fano factor vanishes to 0 as µ 0 for τ = 0 and diverges for any τ > 0 as µ 0. The vanishing → → of F can be understood by noticing from Eqs. (3.33) and (3.34) that I µ and S µ2 for τ = 0 ∝ neq ∝ as µ 0. Physical speaking, this points to the fact that at τ = 0 excess spin noise cannot depend → on which chain is biased, unlike spin current which must. Ultimately the spin Fano factor vanishes like F µ in the absence of temperature bias. ∝ The divergence of F for τ > 0 can be seen formally in Eq. (3.33), where the effects of thermal and chemical biases completely factorize and µ appears only inside the sine prefactor. The spin current I decreases to zero as µ 0 for any τ while the excess noise S remains finite for all µ → neq and τ > 0 [see Eq. (3.34)], so F diverges as µ 0 for all τ > 0. Within the gaussian Luttinger → model description for the QSCs used here, a thermal bias alone does not generate a net spin current

net and reported super-Poissonian shot noise resulting from dipolar interactions. CHAPTER 3. 1D SPIN CHAINS 41

across the chains, and a nonzero chemical bias µ > 0 is required for a net spin ﬂow between the chains. Interpreted physically, what we ﬁnd is the exposure of the underlying fermionic statistics of quasiparticle exchange across the weak link. The exchange coupling, Eq. (3.2), represents hopping of Jordan-Wigner fermions. When µ = 0, as the fermionic spectrum is linearized and the density of states is thus constant at the fermi points, thermal bias on one edge leads to no net current across the weak link whereas excess noise still arises. This physical interpretation is represented in Fig. 3.4 as divergence of the Fano factor for τ > 0 as µ 0. → In Sec. 3.2.2, we noted an exact analogy between the current problem and the problem of stochastic tunneling of electrons between two single fractional quantum Hall edge channels. In the latter problem, the factorization of the thermal and chemical (i.e., voltage) biases in the tunneling charge current has been obtained and is well-known. [78; 92; 93]

3.4.2 Manifestation of Pauli blockade in noise

The xxz quantum antiferromagnetic chains described by Eq. (3.1) can be mapped to a lattice model of interacting fermions via the Jordan-Wigner transformation [69]

   X  z 1   + † Sˆ = cˆ†cˆ j , Sˆ − = cˆ j cos π cˆ†cˆl = Sˆ , (3.45) j j − 2 j  l  j l< j

x y with Sˆ ± = Sˆ iSˆ . This mapping to the fermion model reveals a certain resemblance between the j j ± j spin-1/2 operators and fermions where the absence (presence) of a fermion on site j corresponds to a state with S z = 1/2 (S z = 1/2). In particular, once a spin-1 excitation is injected into the j − j drain chain at j = 0 from the source chain, a second spin-1 excitation cannot be injected into the same site; this leads to a partial blockade of spin transport across the chains, analogous to transport blockade due to Pauli’s exclusion principle observed in electron transport. [94] A direct consequence of this Pauli blockade physics in QSCs can be obtained in the dc noise, the physical origin of which exactly resembles the Pauli blockade picture developed for charge CHAPTER 3. 1D SPIN CHAINS 42

ﬂuctuation suppression between two edge states of a fractional quantum Hall liquid coupled at a quantum point contact. [95] To elaborate on this point, let us now consider a case when both QSCs are chemically biased such that spin currents impinge the junction from both ends in Fig. 3.1. If we denote the spin chemical potentials for the two QSCs as µ1 and µ2, the non-equilibrium dc noise is modiﬁed to

2 2 Z ( " # ) (Jc⊥) a (µ1 µ2)t + + S (µ , µ , T , T ) = dt cos − 1 D− (t)D− (t) . (3.46) neq 1 2 1 2 2π2η2 ~ − 1 2

We now ﬁx the temperatures of both QSCs to the same value, i.e., T = T T , and deﬁne 1 2 ≡ 0

S neq(µ1, µ2, T0, T0) α(µ1, µ2) (3.47) ≡ S neq(µ1, 0, T0, T0) as the ratio of the non-equilibrium noises when both QSCs are chemically biased to that when only one of the QSCs is biased. Fig. 3.5 is then produced by sweeping µ2 from 0 to µ1 while keeping µ1 fixed. In the QSC system (denoted by the solid line), the non-equilibrium noise exhibits suppression as µ µ and it vanishes at µ = µ . This noise reduction can be attributed to Pauli 2 → 1 2 1 blockade, which suppresses the phase space for the scattering of (fermionic) spin-1 excitations at the QSC junction. A starkly contrasting behavior is predicted for the magnon setup. Here, we consider a possibility of both magnon baths having finite chemical potentials µ1 and µ2, and we define the ratio

S m,neq(µ1, µ2, T0, T0) αm(µ1, µ2) , (3.48) ≡ S m,neq(µ1, 0, T0, T0) which again is the ratio of the non-equilibrium noises when both magnon baths are chemically biased to that when only one bath is biased. In the magnon case, introducing µ2 results in more noise, i.e., αm > 1 for µ2 > 0 exhibiting no signature of Pauli blockade, which is a feature unique to fermionic excitations. CHAPTER 3. 1D SPIN CHAINS 43

2 ) ,µ2 (µ1 ↵m 1 ↵( µ1 ,µ 2 )

0 0.5 1 µ2/µ1

Figure 3.5: Excess dc spin current noise as a function of µ2/µ1 for the QSC case (solid line) and the magnon case (dashed line) with T1 = T2 = T0. See the main text for the deﬁnitions of α(µ1, µ2) and αm(µ1, µ2). The suppression obtained in the QSC case is a manifestation of Pauli blockade physics.

3.5 Experimental extraction of spin Fano factor

3.5.1 QSC scenario

The QSC spin Fano factor may be experimentally detected using a Y-junction setup shown in Fig. 3.6. In this setup, the drain QSC is weakly exchange coupled to two source QSCs at its left end and to a metal with strong spin-orbit coupling at its right; the source chains 1 and 2 are chemically biased by spin chemical potentials µ1 and µ2, respectively. We assume that the two source QSCs are identical and that they are exchange coupled with equal strength to the drain chain; departures from this symmetric condition should not aﬀect the following discussion at the qualitative level.

The temperatures of source chain 1, source chain 2, and the drain chain are denoted by T1, T2, and

T0, respectively.

To lowest order in the source-drain coupling, the spin current I at x = x1 reads

I = I(µ1, T1, T0) + I(µ2, T2, T0) , (3.49) CHAPTER 3. 1D SPIN CHAINS 44

µ1

source chain 1

I,S x = x1 T1 drain chain Ic,Sc

detector A T metal T2 0

source chain 2

µ2

Figure 3.6: Depiction of the proposed conﬁguration for extracting the spin Fano factor in the QSC case. Two source chains are individually coupled to a third drain chain at one point. The chains are held at temperatures T1, T2, and T0 respectively, and chemical biases µ1, µ2 in the source chains only. and its dc noise is given by

2 2 Z 2~KkBT0 (Jc⊥) a µ1t + + S (µ , µ , T , T ) = + dt cos D− (t)D− (t) 1 2 1 2 π 2π2η2 ~ 1 0 2 2 Z (Jc⊥) a µ2t + + + dt cos D− (t)D− (t) (3.50) 2π2η2 ~ 2 0 S (0) + S (2)(µ , T , T ) + S (2)(µ , T , T ) . ≡ 1 1 0 2 2 0

Spin current fluctuations in the drain QSC at x = x1 should generate pure spin current fluctuations inside the adjacent metal due to the coupling between the two systems. The latter pure spin current noise should convert into noise in the transverse charge current via the inverse SHE and should be detectable using an ammeter as illustrated in Fig. 3.6. An electrical noise measurement is first made when the two source chains are unbiased, i.e.,

µ1 = µ2 = 0, and T1 = T2 = T0. We denote the electrical noise measured in this conﬁguration with

S c,eq. The charge noise is then measured in a second conﬁguration in which the source chains are CHAPTER 3. 1D SPIN CHAINS 45

asymmetrically biased, i.e., µ = µ , such that I remains zero. We denote the noise measured in 1 − 2 this conﬁguration with S and the excess non-equilibrium noise as S S S . The excess c c,neq ≡ c − c,eq noise in the second conﬁguration should arise solely from the non-equilibrium spin current noise generated at the coupling site. Using the notation of Sec. 3.4.1, this non-equilibrium spin current noise is given by S (µ , T, T ) + S ( µ , T, T ) = 2S (µ , T, T ), where we have assumed T neq 1 0 neq − 1 0 neq 1 0 ≡

T1 = T2 due to the above symmetry condition. It is therefore reasonable to assume that the excess

electrical noise is proportional to the excess spin current noise, i.e., S c,neq = ΘS S neq(µ1, T, T0),

where ΘS is some spin-to-charge noise conversion constant. The temperature of the source chains

T may be elevated above that of the drain chain T0 because the generation of ﬁnite µ1 and µ2 requires charge currents in the injector metals and may cause Joule heating.

Lastly, we turn µ2 off while keeping µ1 at the same value as in the second configuration. There is now a net spin current I(µ1, T, T0) flowing into the drain QSC that will convert into a net charge current via the inverse SHE inside the detector metal. It is reasonable to assume here that this generated charge current is proportional to the generated spin current I = Θ I(µ , T, T ) , where | c| I| 1 0 |

ΘI is some spin-to-charge current conversion constant. Finally, the electrical Fano factor can be computed from the experimental readings, and we ﬁnd that S c,neq Θ S neq(µ1, T, T0) ~Θ F = = S = S F(µ , T, T ) . (3.51) c I Θ I(µ , T, T ) Θ 1 0 | c| I | 1 0 | I

From Fig. 3.4, we know that for any T and T0, the spin Fano factor F approaches 1 as µ1 increases. In the large chemical bias regime where F approaches 1, the ratio of the unknown prefactors can be experimentally extracted via Θ F S = c , (3.52) ΘI ~

thus allowing one to obtain the spin Fano factor F for all µ1 values.

If the temperatures of the source QSCs and the drain QSC are uniform, i.e., T = T0, Fc should display a vanishing behavior as µ 0. However, if Joule heating results in T > T , a thermal 1 → 0 CHAPTER 3. 1D SPIN CHAINS 46

Ic,Sc drain metal Im,Sm interface 1 interface 2 T spin current 0 spin current

µ1,T1 µ2,T2 V1 V2 injector source bath 1 drain bath source bath 2 injector

Figure 3.7: Depiction of the proposed conﬁguration for extracting the spin Fano factor in the magnon case. Two source baths are individually coupled to a third drain bath. The baths are held at temperatures T1, T2, and T0 respectively, and chemical biases µ1, µ2 in the source baths alone.

bias exists in addition to the chemical bias and we expect F to exhibit a diverging behavior shown in Fig. 3.4. Therefore, the spin Fano factor can be used to distinguish between the presence and absence of Joule heating in the source QSC due to charge currents in the injector metals.

3.5.2 Magnon scenario

The spin Fano factor for the magnon scenario can be extracted in a setup similar to Fig. 3.6. We consider the drain magnon bath coupled to two source magnon baths in a T-shaped setup as shown in Fig. 3.7. The injection of spin angular momentum into the source magnon baths are facilitated by charge currents and SHE in the respective injector metals. An important consequence of this

injection process is to generate ﬁnite chemical potentials µ1 and µ2 for the source magnon baths. We assume the symmetry condition (as in the QSC setup), in which the two source magnon baths are identical and they couple to the drain magnon bath with an equal strength. The temperatures

of magnon bath 1, magnon bath 2 and the drain magnon bath are denoted by T1, T2, and T0, respectively.

Electrical noise is ﬁrst measured when the two source magnon baths are unbiased, i.e., µ1 =

µ2 = 0, and T1 = T2 = T0. We denote the electrical noise measured in this conﬁguration with S c,eq.

In the second conﬁguration, the source baths are oppositely biased, i.e., µ1 > 0 and µ2 < CHAPTER 3. 1D SPIN CHAINS 47

0, such that no net magnon spin current is detected at the detector metal, i.e., Im(µ1, T1, T0) = I (µ , T , T ) such that I = 0. We again denote the electrical noise measured in this second − m 2 2 0 m conﬁguration with S and the excess noise as S S S . As in the QSC scenario, the c c,neq ≡ c − c,eq excess noise S c,neq should arise solely from the excess spin current noises that are generated at interfaces 1 and 2 and have propagated to the detector metal. This excess noise in the spin current is given by S m,neq(µ1, T1, T0) + S m,neq(µ2, T2, T0). It is then reasonable to assume that the excess electrical noise measured by the detector metal is proportional to this excess spin current noise, i.e., S S (µ , T , T ) + S (µ , T , T ). The temperature of the source baths may once c,neq ∝ m,neq 1 1 0 m,neq 2 2 0 again be elevated above that of the drain bath T0 because charge currents in the injector metals may cause Joule heating in the magnon baths.

Once µ2 is turned off while keeping µ1 at the same value as above, a net spin current Im(µ1, T1, T0) flows into the drain bath. It is reasonable to assume that the charge current generated at the detector metal obeys I I (µ , T , T ). c ∝ m 1 1 0 Taking the ratio F = S / I , we find that c c,neq | c|

S m,neq(µ1, T1, T0) + S m,neq(µ2, T2, T0) F c ∝ I (µ , T , T ) | m 1 1 0 | S , (µ , T , T ) S , (µ , T , T ) = m neq 1 1 0 + m neq 2 2 0 (3.53) I (µ , T , T ) I (µ , T , T ) | m 1 1 0 | | m 2 2 0 | F , ∝ m where Fm is shown in Fig. 3.4. Here, unlike the QSC case, regardless of the temperatures and relative biases of the magnon baths, Fc should remain constant. Therefore, the contrasting behavior between the QSC and magnon spin Fano factors, as plotted in Fig. 3.4, should be electrically measurable via inverse SHE signals. CHAPTER 3. 1D SPIN CHAINS 48

drain chain drain metal z I I y⇥ x c l ⇥

spin conversion profile d Figure 3.8: Depiction of spin to charge current conversion between a QSC and normal metal contact. We extract an approximation for the size of the conversion eﬀect between the two materials from this geometry. Here, l is the thickness of the interface and d is the width of the metal contact.

3.5.3 Estimate of noise signal strength

We derive an estimate for the magnitude of the noise eﬀect measurable via conversion in a strongly spin-orbit coupled metal spin drain (e.g., Pt). Fig. 3.8 shows the geometry of the contact, which is standard in recent two terminal spin chain experiments, for instance in Ref. [19] (using spin drain chain material Sr2CuO3). If we consider N QSCs attached laterally per unit area, we may write the total spin current density impinging on the normal metal as NI(t), where I(t) is the spin current due to a single QSC. For a normal metal of thickness d and spin diﬀusion length λ, we may write

d x d the spin current density proﬁle across the normal metal as js(x, t) = NI(t) sinh −λ / sinh λ , [96] assuming that the spin current I(t) [see Eq. (3.33)] arriving at the interface fully penetrates into the metal and a boundary condition of vanishing spin current at the outer edge of the sink material. Modeling spin to charge conversion via the SHE, characterized by the spin Hall angle Θ, we derive

2e the charge current density in the normal metal as jc(x, t) = Θ ~ js(x, t). We then ﬁnd the total charge current ﬂowing in the metal by integrating over the cross-section

! 2e d I (t) = Θ NlλI(t) tanh , (3.54) c ~ 2λ CHAPTER 3. 1D SPIN CHAINS 49

where l is the height of the interface area (see Fig. 3.8), and characterize the associated dc charge R noise S = dt I (t)I (0) as c h c c i

" !#2 2e d S = Θ Nlλ tanh S. (3.55) c ~ 2λ

We determine N from the documented values for the lattice spacings of Sr2CuO3, [97] use Θ = 0.1 and take λ = 2 nm, d = 7 nm, and l = 1 mm. Using the known properties of Pt, we estimate

16 2 the magnitude of the ﬂuctuations as S 10− V s in the given conﬁguration. This calculation c ∼ assumes a temperature of T = 20 K, again following Ref. [19], which results in background

14 2 Johnson-Nyquist noise of S 10− V s in the contact. Voltage noise measurements of order JN ∼ 20 2 10− V s have been reported in, e.g., Ref. [63]. We thus believe the Fano factor results of Sec. 3.4.1 should be measurable with existing equipment and techniques.

3.6 Conclusion

In this chapter, we considered two spin systems: one comprised of two semi-infinite QSCs that generate noise across a weak coupling, and a second comprised of two semi-infinite magnon baths that also generate noise across a weak coupling. In either case, we have derived the bulk spin current and bulk spin current noise at a point close to a measurement metal contact with strong spin-orbit coupling. Our analysis shows that it should be possible to differentiate the systems using a quantity known as the spin Fano factor, where Pauli blocking and concomitant current suppression in the QSC case result in dramatically different behavior than for the same quantity in the magnon scenario. Additionally, we show that Pauli blocking has signatures directly accessible in the noise. These results exhibit the fermionic nature of the spin-1/2 operator in the QSC. We then proposed an experimental method by which to extricate the spin Fano factor and therefore experimentally compare the two systems we consider. In so doing we assumed propor- CHAPTER 3. 1D SPIN CHAINS 50 tional relationships between the excess spin current noise and the excess charge noise, and the spin current and charge current in the metallic contacts, i.e., both S c,neq = ΘS S neq(µ1, T, T0) and I = Θ I(µ , T, T ) . However, it would be desirable to develop a microscopic model for the | c| I| 1 0 | conversion between these quantities thereby elucidating the regime of validity of the assumption and providing microscopic determinations of ΘS and ΘI. In the absence of such a model, we may consider other methods by which to measure these quantities. This is the work of the next chapter. Chapter 4

Measuring spin shot noise with an LC resonator

In this chapter, we continue our examination of coupled 1d quantum paramagnets. In particular, we propose a technique for detecting spin current noise generated across a junction between two 1d quantum paramagnets — arranged as discussed in the previous chapter — using a high-quality microwave resonator coupled to a transmission line which is impedance matched to a downstream photon detector. Photons in the microwave resonator couple inductively to the spins in the spin subsystem, and the noise created at the junction imprints itself into the output photons propagating along the transmission line. We utilize the input-output formalism to describe light propagating along the coupled transmission line, and show that the proposed system is capable of extracting both the dc and ﬁnite frequency noise via the output photon ﬂux. This result should be widely applicable, in addition to addressing an aspect of the detection issue mentioned in the conclusion of the previous chapter.

51 CHAPTER 4. QSC SPIN NOISE MEASUREMENT 52 4.1 Introduction

Shot noise in mesoscopic conductors arises as a consequence of the quantized nature of charge transport [98]. Its utility far exceeds that of the equilibrium (Johnson-Nyquist) counterpart, as shot noise can be used to extract the charge [77] and the statistics of relevant charge carriers as well as to probe quantum many-body effects [99] and entanglement [100]. In insulators, charge fluctuations are gapped out but localized electron spins can still generate fluctuations in pure spin current, or spin current noise. It is then anticipated that these pure spin currents can reveal nontrivial properties of the underlying spin system in analogy with the above-mentioned charge scenario. Indeed, recent theoretical inquiries on pure spin current noise in insulating magnets have purported to show its utility in revealing the quantum uncertainty associated with magnon eigenstates [64], the non-trivial spin scattering and heating processes generated at a detector interface [101] and the effective spin and statistics of the tunneling spin quasiparticles [49]. Insulating materials present novel concerns. In metallic systems, spin current noise detection is possible via its observable effects on simultaneous charge current fluctuations [102; 55; 59; 60]; similar methods are clearly not available for insulators. The prevailing spin current detection method in the latter has been the inverse spin Hall effect wherein spin current is detected electrically by coupling a metal with strong spin-orbit interactions to the active spin system [23]. However, this spin Hall detection scheme, while reliable for spin current detection, may be un- reliable for the measurement of spin current noise because spin Hall conversion processes can result in noise enhancement [1]. Other techniques exist for detecting spin fluctuations including spin noise spectroscopy [103], SQUID-based spectroscopy [104], and quantum-impurity relaxometry [105; 106; 107]. However, spin noise spectroscopy does not measure the current-current correlator, i.e., spin current noise, and neither SQUID-based approaches nor relaxometry are amenable to extracting current-current correlations in the spin sector either. Therefore, in connecting theory to experiment, a measurement technique capable of directly detecting spin current noise is of CHAPTER 4. QSC SPIN NOISE MEASUREMENT 53

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t =0 3nm lumped ⇡ 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0 -1 -0.5 0 0.5 1 resonator 10-5 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x = x1 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 x = x2 spin spin ⇠c sink

spin spin chain 1 spin chain 2 injector

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z xAAAV4HicjVhLb9tGEGbSV6o+krT1qReigo30EljO+xY7MiTk7ch2gkiGsSRXIqPlw7sr2bLCe4EeigLtpf+oP6H/prMk9eAMpVaAKGq+bz7Ozs4Ol3QSESi9vf3PlauffPrZ519c+7L21dfffHv9xs3vjlU8ki4/cmMRy3cOU1wEET/SgRb8XSI5Cx3B3zrDJwZ/O+ZSBXF0qCcJPwnZIAr6gcs0mA4uTm/Ut29vZx+bnjSKk7pVfF6f3vzh754Xu6OQR9oVTKluYzvRJ1MmdeAKntZ6I8UT5g7ZgHfhNGIhVyfTLNLU3gSLZ/djCd9I25l12WPKQqUmoQPMkGlfYcwY59jmMhjxc32RweUIdP/hyTSIkpHmkZsH0B8JW8e2SYftBZK7WkzghLkygDHYrs8kczUkrVbbPGzv2532qzeHT44OO7XNluR8aDOR+MzhutaDq7pxGLLIm/aYSM0BoLQMODw1B43tAzAPGPxBds/wPS6Ig7HzRAUijhByCcglvYJvHIhV+0FqjhQJ4tQciH3IUnNIEgwIAwgoNw8jzw3yvAoJzdXDEbJGYIywMQFjEiCjNEFKP0ZmBVYVDEg2J2AfVSatb9R9LH8BVpdYz8Eah3yA1d+C/VWFvWnmqlkxhy2wt4pJl3wZ6QDSqRhAc4WSNhnWDOdsH6zvAqJu0gBZSKGmzSrhcIAmIOyCoFDiPUM3axqqGkNcpt2dE8gfLJM5y6430qU/Oykp6mUvL6N7FTwZjyIv7TaAKXhf3wKi3ZPBwNc/kzGpsxGTfEHuLsgnSNYdSTFZMHvTeiMn9nAAzFEL3seF4kesOOZuPqIsZLPCB0EEfVAyyDW0GuPZgxHaPQ5wbscXMyI67d5ZqQJh9uo78L3zHyoQ9P3/o9Kr34XvPfjeX6MI1ZF2787mAMkZva1cbgvUFiqLeUJaIPZoVXAzua08uK36vS0IrVd/sFV/uFV/tCZIuX4GZmGuF+jn41wR2iK28kirZmCdEp6B9TGtm0sXR1UkbLUiiwZLhS/gr+CLsu7JzIBXa5UTEij5rxNzJKsKoHpZDbkuL791yrFgSV4BldKZ9Xa2BNeIQH1yka/BSpXMv4jmzlolh5+ls/ni0JvMFgvPBs84ZrZWMfZKKlExqajJL6lUM5xQzFXCkdBmg0hiyThGZcbYIf3Y4Wyuw0QwoAOKWCFSCTsqmfsr2NFq4p8RjH8l7MXnZmtgfmCc8TmCR0l2e59B5fuD6Q9QJLD/zSatyl8sOGaKqyhRtjWJo1HowD0MrVbH3IMD4fGqDZ5uztGqG7irXLNBC534Ypqd4x0NX+DZOVlX6RSOsfDMljkWtiSMFxmjb78gSKfsO+2QvuEoVXgrAiXoygmZVbON8pkusS5okY4UUjI7w1OFE5kT81xSPnXgk4rLkzXCLytYlyTIAIUYEMYHxPhAGE8R4ylhjBFjTJOFGCFhDBFjSBkM5xssmHSGOGeEsYsYu4Sxjxj7hOEjBq3vA8Q4IIw9xNgjDAcxHFrmWahmUwz74ek+WQd+K68Su4UBeNyYFVD+5AGEJoenYslfgNyrhEumYwnbetMIZldorqRpWNCHchWqoLlOzWEFDk+wEE7THFGPHKelpU7uA6q1FF6LoO0CVa6ctgm6u4SSMnDbS8rE128X6SOI0y6HTF07hWuHIE6HruqK9uY7qoKnaBvs91Gr7FOtJHukzsTGTJp/WOWiPCDaCqMygd5IJ2UC6Wa+ebwrgjCnGD4uwGOCRAVCrum/LJCXGBmb51sYavXrCAd19aq0atSJ8mdZ2hY5pi1dlcR7VMR7RJCnBUIar88KhPRBP3ujk+Uzf61DKqiASU/x9wqE9qNLnr2/WBrQNvF+U3i/IchugZCF5r8vkPcY0Z35jZOsFT3fvtAxOExmJclkRS0y2S+wXkWx62dz2WcEy5Kag+aVGYajBWo2VBgO/DlO7hd63hQW48Vl20/nK7RWO71Rb+BXnfTkeOd2A84P7tYfPyxeg16zfrR+sm5ZDeuB9dhqW6+tI8u1uPWr9Yf154az8cvGbxu/59SrVwqf763SZ+OvfwHjSXlq

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A microwave resonator, comprised of a loop of inductance L and a capacitor of capacitance C, is set in the xy plane a distance δ from spin chain 2, which is coupled with strength ξc to spin chain 1. unique interest. In this chapter, we will show that a microwave resonator circuit can be used to directly measure nonequilibrium pure spin current noise generated in a quantum magnet. Here, we illustrate this possibility in the context of a one-dimensional quantum antiferromagnet chain. As shown in Fig. 4.1, we consider a situation in which one spin chain (chain 1) is driven out of equilibrium by spin injection at its open end with the downstream end weakly exchange-coupled to a second spin chain (chain 2) set near a microwave cavity. The microwave cavity couples inductively to chain 2, and measurements are transmitted electrically into a transmission line where the photon number flux encodes spin current fluctuations across the coupled spin chain subsystem. We show that by coupling spins to light it is possible to measure the junction spin current noise without resort to the inverse spin Hall effect. This chapter is arranged as follows: in Sec. 4.2 we introduce the workings of the system heuris- tically, and then develop the miscroscopic model in Sec. 4.3. In Sec. 4.4 we utilize the model developed to examine the ac and dc spin-current noise spectra of noise generated across the chain-to- chain coupling and imprinted in the output photons along the transmission line, and therefore show how extraction of the spin Fano factor is possible. Finally, we conclude the chapter in Sec. 4.5. CHAPTER 4. QSC SPIN NOISE MEASUREMENT 54 4.2 Heuristic picture

We first qualitatively illustrate the mode of operation. Let us consider spin current tunneling between two coupled quantum antiferromagnet chains with uniaxial spin symmetry along the z axis (see Fig. 4.1). Spin injection, facilitated by, e.g., the spin Hall effect at the upstream end of chain 1, establishes a nonequilibrium spin bias between the two chains, and the resulting spin current flowing across the chains can be absorbed and measured at the right metal reservoir via the inverse spin Hall effect. A rectangular wire loop with inductance L is placed a distance δ away from chain 2, and is oriented so that it lies in the xy plane and its bottom edge stretches from x = x1 to x = x2. In this geometry, the magnetic flux through the wire loop sharply increases by one unit when a single z-polarized spin-1 quasiparticle enters the “influence region” defined by x1 < x < x2 on chain 2. The magnitude of that unit depends on the distance δ and the width w of the loop. Considering the spin-1 quasiparticle as a magnetic dipole m = γ~ˆz located at position x on chain 2 (γ being the − gyromagnetic ratio), the flux through the loop reads

Z s Z w µ0mz 2 2 3/2 Φ(x) = − [(x0 x) + (y + δ) ]− dx0dy, (4.1) 4π 0 0 −

where s = x x is the influence region. Fig. 4.2 shows a sharp increase in the flux as the 2 − 1 quasiparticle tunnels into the influence region for various δ. We find that the flux is essentially independent of the quasiparticle position in the influence region and that the dipolar fields from spins located outside of the influence region are effectively irrelevant to the total flux through the loop.

Imagine now that the x = x1 edge of the wire loop is located at site i1 on chain 2 and the x = x2

edge at site i2. Then the term in the Hamiltonian modeling the tunneling of spin-1 quasiparticles

from site i1 to i1 + 1 on chain 2 (allowed by the intrachain exchange interaction) involves a term of

+ + the form S − S (S S − ), where S ± denotes the usual spin raising (lowering) operator on 2,i1 2,i1+1 2,i1 2,i1+1 ν,i CHAPTER 4. QSC SPIN NOISE MEASUREMENT 55

109 2.2 ⇤ ) 42

21 1nm ⇡ 10 ⇥ ( 2 21 2nm Tm

⇥ ⇡ ) 3nm x

( ⇡ 0 -1 -0.5 0 0.5 1 x x -5 1 x 2 10 Figure 4.2: A plot depicting magnetic flux through the loop as spins enter spin chain 2. The sharp drop-offs indicate that spins outside of the “influence region” x1 < x < x2 are effectively ignored by the microwave resonator.

chain ν at site i. However, the fact that every tunneling process is accompanied by a ﬂux change Φ

+ iΦq /~ in the loop requires that the tunneling operator is modiﬁed to S S − e 0 + h.c., where q is 2,i1 2,i1+1 0

an operator obeying [q0, φ0] = i~ and translates the inﬂuence ﬂux φ0 through the loop by Φ, i.e.,

iΦq /~ iΦq /~ e± 0 φ e∓ 0 = φ Φ. (4.2) 0 0 ∓

In this conﬁguration, q0 is the charge on the capacitor C (see Fig. 4.1), and φ0 and q0 form a conjugate variable pair. The spin tunneling operator endowed by the ﬂux translation operator indicates that the spin tunneling process involves interactions with the electromagnetic environment formed

by the loop-capacitor subsystem. In a similar fashion, the spin tunneling term at the x = x2 edge is

+ iΦq /~ also modified to S S − e− 0 + h.c. . 2,i2 2,i2+1 Energetic considerations show that the wire loop in principle affects spin transport in chain 2. That is, for every unit of flux Φ tunneling into the influence region, the energy of the inductive

2 system increases by EΦ = Φ /2L, where L is the inductance of the resonator. As a result, two regimes emerge: what we call the noninvasive and invasive regimes. In the noninvasive regime,

EΦ is much smaller than the non-equilibrium bias µ, the temperature T, and the photon frequency Ω, i.e., E µ, k T, ~Ω (note that we will define the photon frequency later). In this regime, Φ B we may focus exclusively on the effect of nonequilibrium spin transport on the electromagnetic CHAPTER 4. QSC SPIN NOISE MEASUREMENT 56 environment and neglect the back-action of the environment on the spin subsystem. A fluctuating spin current at the junction leads to a fluctuating magnetic flux through the wire loop and thus to a fluctuating electromotive force inside the resonator by Faraday’s law of induction. As mentioned previously, this fluctuating electrical signal is ultimately detected in the transmission line via an output photon number flux containing a direct imprint of the spin current noise. In the invasive regime, the environmental effect is not negligible and the spin transport in chain

2 should deviate from their unperturbed values. For EΦ much greater than the nonequilibrium spin bias µ and temperature T, i.e., µ, k T E , tunneling events become increasingly unfavorable B Φ energetically due to the resistive electromotive force emerging from Lenz’s law and leads to a suppression in the spin transmission through the chain. We refer to this phenomenon as inductive blockade, which may be thought of as the magnetic analog of the well-known Coulomb blockade studied extensively in quantum conductors [108; 109]. If the junction resistance is strong enough to suppress elastic scattering between the nodes, tunneling quasiparticles must have suﬃcient energy to excite environmental modes and proceed inelastically. However, the regime may be challenging to realize in practice due to the weakness of the spin-light interaction. In the remainder of the work, we focus on the noninvasive regime and present the technical calculations to establish the above heuristic results.

4.3 Microscopic theory

4.3.1 The Model

We consider two identical semi-inﬁnite xxz quantum antiferromagnet chains coupled together at their ﬁnite ends, one additionally coupled inductively to a microwave resonator and the resonator itself placed in series with a transmission line on which measurements are performed. The spin CHAPTER 4. QSC SPIN NOISE MEASUREMENT 57

chains are modeled by the usual xxz Hamiltonian

X∞ h i 1 + iΦq0/~ iΦq0/~ z z Hν = J 2 S ν, jS ν,− j+1 1 + δ j,i1 δν,2e + δ j,i2 δν,2e− + h.c. + ∆S ν, jS ν, j+1 , (4.3) j=0

where Sν, j is the spin-1/2 operator on chain ν at site j, J is the intra-chain exchange scale and

δi, j is the kronecker delta. The exponential factors encode the coupling of spin chain 2 to the electromagnetic environment, as discussed previously. We assume throughout that the spin chains are in the so-called gapless phase, i.e., ∆ < 1, where it can be suitably described using the | | Luttinger liquid formalism [11]. We model the transmission line as an inﬁnite array of parallel LC resonators with lineic capacitance c, lineic inductance l, and characteristic impedance z = √l/c. Its Hamiltonian reads

Z ( 2 2 ) ∞ φ (xt) [∂xt q(xt)] HTL = dxt + , (4.4) 0 2l 2c

where xt labels the position along the transmission line, q(xt) and φ(xt) denote the local charge and

ﬂux, respectively, and [q(x ), φ(x0)] = i~δ(x x0). Located at the end of the transmission line, i.e., t t t − t at xt = 0 (see Fig. 4.1), is the lumped series LC resonator with capacitance C and inductance L, and governed by the Hamiltonian φ2 q2 H = 0 + 0 , (4.5) r 2L 2C

where q q(x = 0) and φ φ(x = 0). The total Hamiltonian for the full system then reads 0 ≡ t 0 ≡ t P H = ν Hν + Hr + HTL + V(t), where

1 + iµt/~ V(t) = J (S S − e + h.c.) (4.6) − 2 c 1,0 2,0 describes the tunneling of spin-1 quasiparticles across the spin chains allowed by the weak inter-

iµt/~ chain exchange interaction Jc. The oscillatory factor e captures the fact that the spin chemical CHAPTER 4. QSC SPIN NOISE MEASUREMENT 58

potential in chain 1 has been raised to µ > 0 via spin injection at its upstream end.

4.3.2 Input-Output Formalism

The standard input-output approach [110] proceeds by ﬁrst expanding the local charge q(xt, t) in Fourier series

Z r ∞ ω ~ h i d iω(xt/v t) iω(xt/v+t) q(xt, t) = ao(ω)e − + ai(ω)e− + h.c. , (4.7) 0 2π 2ωz

1/2 where v = (lc)− and ai,o are the incoming and outgoing photon ﬁelds on the transmission line.

Noting that the Heisenberg equations of motion evaluated at the lumped resonator xt = 0 give

φ0(t) = Lq˙0(t) (4.8) ∂xq(xt 0, t) q0(t) Φ + φ˙ = → [I˜ (t) I˜−(t)] , 0 c − C − ~ i1 − i2

one can solve for the output photon ﬁeld in terms of the known input photons,

r ˜+ ˜ Γ∗(ω) Φ 2zω I (ω) I−(ω) a (ω) = a (ω) + i1 − i2 . (4.9) o − Γ(ω) i ~L ~ Γ(ω)

Here,

˜ + iΦq0/~ Ii±(t) = i(J/2)S i S i−+1e± + h.c. (4.10)

is the operator for the bulk spin current in chain 2 flowing to the right at site i (the exponential factor encodes the effect of the electromagnetic environment), Γ(ω) = ω2 Ω2 + iκω, κ = z/L − 1/2 is the rate at which photons decay into the transmission line, and Ω = (LC)− is the resonance frequency. The first term in Eq. (4.9) then corresponds to the reflection of incoming photons while the second term describes the emission or absorption of additional photons caused by the tunneling of CHAPTER 4. QSC SPIN NOISE MEASUREMENT 59

spin-1 quasiparticles into and out of the inﬂuence region. The incoming photons are assumed to

be equilibrated at resonator temperature Tt, which may be distinct from temperature T of the spin chains, and obey

a†(ω)a (ω0) = 2πn (ω)δ(ω ω0) , (4.11) h i i i0 B −

~ω/k T 1 where n (ω) = (e B t 1)− is the Bose-Einstein distribution describing the thermal photons in B − the transmission line. As discussed previously, noninvasive detection of the spin current noise is conducted in the limit where spin-photon coupling strength quantified by Φ remains sufficiently small so as to leave the spin chain subsystem approximately unperturbed while the coupling of the resonator to chain 2 is simultaneously kept strong enough for detection. Solving for the output photon flux using Eq. (4.9) is difficult, in principle, because the photon field itself enters the spin current operator ˜ Ii±(t) through q0(t) and this calls for self-consistency. However, since the correction to the output photon flux arising from the spin subsystem is anticipated to be suppressed by an overall multi- plicative factor proportional to Φ2, c.f. second term in Eq. (4.9), the noninvasive assumption allows us to set the spin-photon coupling to zero, i.e., Φ = 0, when computing spin transport quantities, thus effectively lifting the self-consistency requirement.

4.3.3 The Output Photon Flux Spectrum

With this in mind, let us now examine the output photon ﬂux spectrum a†(ω)a (ω0) by insert- h o o i ing Eq. (4.9) into the expectation value while invoking the noninvasive approximation mentioned above. The output photon ﬂux spectrum then reads

2EΦκ~ω a†(ω)a (ω) = n (ω, T ) + [(1 + n (ω, T ))(S (ω, µ, T) + S (ω, µ, T)) h o o i B t ~4 Γ(ω) 2 B t i1 i2 | | n (ω, T )(S ( ω, µ, T) + S ( ω, µ, T))] , (4.12) − B t i1 − i2 − CHAPTER 4. QSC SPIN NOISE MEASUREMENT 60

R iωt where S (ω, µ, T) = dte− I (t)I (0) denotes the spin current noise at site i on chain 2; we i h i i i

remind the reader that the transmission line (detector) temperature Tt is distinguished from the

spin temperature T. Here, the local spin current operator I (t) I˜±(t, Φ = 0) is now deﬁned in the i ≡ i absence of the electromagnetic environment and the expectation values are taken by setting Φ = 0

in H2 [see Eq. (6.1)]. In obtaining Eq. (4.12), we also ignored cross-correlations between spin

current fluctuations at x1 and x2. This is a good approximation, because nonlocal spin correlations decay exponentially with distance at finite temperature. We note that for T = 4 K Eq. (4.18) gives an estimate of the length scale of nonlocal correlations, and we find that the correlations are suppressed on the order of x x > 10 nm. | 2 − 1| Recall that in the gapless phase ∆ < 1, each semi-infinite spin chain can be described as a | | chiral Luttinger liquid [49; 11]

Z ~u ∞ 2 Hν = dx [∂ ϕν(x)] , (4.13) 4πK x −∞ where the chiral boson ﬁeld ϕν(x) obeys

[ϕ (x), ϕ (x0)] = iπK sgn(x x0) , (4.14) ν ν − the boson speed and the Luttinger parameter, respectively [85], are given by

πJa √1 ∆2 = u −1 (4.15) 2~ cos− ∆ 1 K = , (4.16) 2 (2/π) cos 1 ∆ − −

and a is the lattice constant. In this case, spin current inside chain 2 is essentially carried by (ballistic) noninteracting bosonic modes so the spin current ﬂuctuations produced at the junction

remain unmodiﬁed as they propagate downstream to sites i1 and i2 [49; 111]. Therefore, the local

spin current noise at any point in the bulk of chain 2, i.e., S i(ω, µ), is given by the background CHAPTER 4. QSC SPIN NOISE MEASUREMENT 61

Johnson-Nyquist noise S 0(T) plus the noise generated at the tunnel junction between the two spin chains dS (ω, µ, T). The total spin current noise at any site i is then given by S i(ω, µ, T) = S 0(T) + dS (ω, µ, T), where [49]

Z 2 ∞ µt iωt dS (ω, µ, T) = 2ξ dt cos D(t, T) e− , (4.17) c ~ −∞ with  2/K  πkBTη   u~  D(t, T) =  h i , (4.18)  πkBT  sin u~ (iut + η) 

1 ξ = J a/2πη, η k− is the short distance cutoﬀ of the theory, and k is the Fermi wavevec- c c ∼ F F tor [112].

4.4 Noise detection

4.4.1 AC Noise Detection

Equation (4.12) shows that a single mode electromagnetic environment can imprint transport quantities generated at a junction coupling two quantum magnets on the number of total output photons propagating along an attached transmission line. Reference [113] provides an example of a single microwave photon detector based on a superconducting ﬂux qubit capable of detecting individual photons propagating through a transmission line with a refresh rate of 400 ns, a narrow band- ∼ width, a well-characterized eﬃciency of 0.66 0.06, and a low dark count rate of 0.01. If ∼ ± ∼ such a detector with bandwidth B is attached at the end of the transmission line in Fig. 4.1 and is designed to count propagating microwave photons with a central frequency Ω, Eq. (4.12) gives the expected rate of photons arriving at the detector as

E Σ(Ω, µ) N(Ω) = n (Ω, T )B + Φ , (4.19) B t ~Ω ~2 CHAPTER 4. QSC SPIN NOISE MEASUREMENT 62

100100

8080

60 low temperature

) 60 regime ⌦ (

N 4040

2020

00 0 0.0251 0.205 0.0753 0.41 10-5 kBTt/h⌦

Figure 4.3: A plot of N(Ω) as a function of transmission line temperature Tt. Here, we use Ω/2π = 1 GHz and spin temperature T = 4 K to remain in the dc spin transport regime, and spin bias µ = 0.03J. The plateau in the limit of small Tt (shaded in blue) is the dc spin current noise. N(Ω) exhibits the same qualitative behavior for essentially all spin biases µ, thus allowing one to quantify the noise using this extraction method for various values of µ. where Σ(Ω, µ) [1 + n (Ω, T )]dS (Ω, µ, T) n (Ω, T )dS ( Ω, µ, T) . (4.20) ≡ B t − B t −

Here, we have assumed a high-quality resonator obeying κ . B Ω. The ﬁrst term in N(Ω) is the number of background thermal photons while the second term represents photon emission and absorption during interchain tunneling. In the limit of low transmission line/detector temperatures k T ~Ω, we have n (Ω, T ) 1 and thus obtain B t B t

E dS (Ω, µ, T) N(Ω) Φ . (4.21) ≈ ~Ω ~2

This is a central result of this chapter, i.e., the output photon ﬂux gives a direct measurement of the ac spin current noise generated at the quantum magnet junction. CHAPTER 4. QSC SPIN NOISE MEASUREMENT 63

4.4.2 The DC Limit and the Fano Factor

If the spin temperature is held higher than the resonance frequency, i.e., ~Ω k T, the output B photon ﬂux directly measures the dc component of the spin current noise, i.e.,

E dS (Ω = 0, µ, T) N Φ . (4.22) ≈ ~Ω ~2

Since the spin current I across the junction can be measured directly using inverse spin Hall eﬀect at the right metallic reservoir (see Fig. 4.1), the proposed system allows one to extract a quantity directly proportional to the dc spin Fano factor, deﬁned as the noise normalized by the current

dS (Ω = 0, µ, T) F = , (4.23) ~I

recently studied in Refs. [64; 101; 49]. Figure 4.3 shows a plot of photon flux N(Ω), Eq. (4.19), as a function of the detector temperature Tt. It exhibits the same qualitative behavior for essentially all spin biases µ, thus allowing one to quantify the noise using this proposed extraction method for various values of µ. The plot in the figure is generated for a spin tunnel junction with coupling strength ξ 0.01J and spin bias c ∼ µ = 0.03J, intrachain exchange scale of J/k 103 K [19], spin temperature of T = 4 K , and an B ∼ inductor loop of inductance L 1 nH with dimensions x x = w = 10 µm placed δ 1 nm from ∼ 2 − 1 ∼ 11 spin chain 2. Under these conditions, we estimate E 10− K 2 Hz. For these parameters Φ ∼ ∼ and at resonance frequency Ω/2π = 1 GHz 1, we expect the chain-to-cavity interaction to produce approximately 26 photons per second in the low temperature regime (shaded in blue), where the photon flux converges to the quantity in Eq. (4.21). We believe the detection of this number of output photons is within the reported capabilities of a single microwave photon detector impedance matched to a transmission line [113].

1Here, we assumed a cut oﬀ length in the Luttinger theory of η/a 10 ≈ CHAPTER 4. QSC SPIN NOISE MEASUREMENT 64 4.5 Conclusion

In this chapter, we have addressed the measurement issue encountered in the previous chapter. In particular, we have shown that by placing a high-quality microwave resonator circuit in close proximity to a pair of exchange-coupled spin chains it is possible to directly measure spin transport quantities at a junction between the two 1d spin systems. When the electromagnetic environment interacts weakly with the spin system, the setup allows measurement of both finite frequency and dc spin current fluctuations by examining the total number of output photons produced by interactions with the environment. This work opens doors to the possibilities of exploring the photodetection statistics of radiation produced at a junction between two quantum magnets and the generation of antibunched photons by exploiting the similarity between the current spin subsystem and a tunnel junction between two quantum conductors [114; 115]. Our theory can also be extended to describe spin transport between tunnel-coupled gapped spin systems (e.g., quantum Ising chains) that may mimic the dynamical Coulomb blockade physics of superconducting Josephson junctions coupled to electromagnetic environments [109]. This concludes the section of this thesis examining 1d quantum paramagnets, and from here we move into an examination of effectively 2d quantum paramagnets. Chapter 5

Measuring QSL density of states using spin ﬂuctuations

In this chapter, we propose an experimental method utilizing a strongly spin-orbit coupled metal to quantum magnet bilayer that will probe quantum magnets lacking long range magnetic order, e.g., quantum spin liquids, via examination of the voltage noise spectrum in the metal layer. The bilayer is held in thermal and chemical equilibrium, and spin fluctuations arising across the single interface are converted into voltage fluctuations in the metal as a result of the inverse spin Hall effect. We develop the theoretical workings of the proposed bilayer system, and provide precise predictions for the frequency characteristics of the enhancement to the ac electrical resistance measured in the metal layer for three candidate quantum spin liquid models. Application to the Heisenberg spin- 1/2 kagome´ lattice model should allow for the extraction of any spinon gap present. A quantum spin liquid consisting of fermionic spinons coupled to a U(1) gauge field should cause subdominant Ω4/3 scaling of the resistance of the coupled metal. Finally, if the magnet is well-captured by the Kitaev model in the gapless spin liquid phase, then the proposed bilayer can extract the two-flux gap which arises in spite of the gapless spectrum of the fermions. We therefore show that spectral analysis of the ac resistance in the metal in a single interface, equilibrium bilayer can test the

65 CHAPTER 5. SPIN NOISE AND QSLS 66 relevance of a quantum spin liquid model to a given candidate material.

5.1 Introduction

Quantum spin liquids (QSLs) refer to intriguing states of quantum spin systems in which strong quantum fluctuations prevent spins from ordering down to zero temperature and the prototypical wavefunction exhibits extensive many-body entanglement [5; 6]. They are endowed with fascinating physical properties like non-local excitations [116; 117] and non-trivial topology [118; 119], and substantial pioneering work has been accomplished in the pursuit of a physical instantiation of this long-sought-after phase of matter. The most promising candidates to date include the mineral herbertsmithite [14], certain organic salt compounds [15], and the so-called Kitaev materials [16], and their experimental studies have ranged from nuclear magnetic resonance [120; 121; 122; 123], to susceptibility and heat capacity investigations [124; 125; 126], to thermal transport studies [127; 128; 129; 130; 131; 132], and to neutron scattering [133; 134; 135; 136]. Theoretical works have shown that QSL ground states are realized in the antiferromagnetic S = 1/2 Heisenberg model on the kagome´ lattice [7; 137; 138; 139; 140], in models that couple low-energy fermionic spin excitations to an emergent U(1) gauge field [141; 8; 142; 143; 144; 145], as well as in the exactly solvable Kitaev model [9] in two [146; 4] and three [147; 148] dimensions. However, while the number and types of potential QSL models have proliferated, the experimental methods available for studying them have not [149]. It is therefore important to the search for compounds possessing QSL ground states to propose previously unknown techniques by which to probe and categorize candidate materials. The inverse spin Hall effect (ISHE) [24] is an essential component of the spintronics reper- toire [23], and provides an opportunity to develop probes of unconventional quantum magnets. This phenomenon has found utility in detecting spin currents induced by thermal gradients [150; 151; 19] and in performing non-local spin transport experiments through magnetic insulators [17; CHAPTER 5. SPIN NOISE AND QSLS 67

18; 67; 68; 152]. The prime function of the ISHE is that it acts as a transducer between spin and charge current densities in metals with strong spin-orbit coupling. One may therefore envisage a bilayer system in which a strongly spin-orbit coupled heavy-element metal (e.g., Pt, Ta, W) is deposited on top of a quantum magnet, i.e., consider coupling the metal to a spin dissipating subsystem that freezes out charge degrees of freedom to focus solely on the spin sector, while holding the entire bilayer in thermal and chemical equilibrium. The addition of a spin dissipating subsystem will result in equilibrium spin current fluctuations across the interface [153; 154], which, as a result of the ISHE, are converted into charge fluctuations inside the metal — charge fluctuations that may encode information about the microscopic structure of the quantum magnet. The QSL-to- normal-metal bilayer therefore presents a table-top setup that can be used to electrically probe the low energy density of states of the QSL material. This type of bilayer system has garnered some study with respect to ferromagnetic insulators hosting magnons [155]; however, the utility of the ISHE as a noise conversion mechanism has not yet been explored in the context of exotic quantum magnets in general and the search for QSLs in particular. In this chapter, we propose a relatively simple mechanism that can probe exotic 2d QSL ground states possessing gauge fluctuations and extensive many-body entanglement via equilibrium or near-equilibrium measurements. We will develop the theory underpinning the aforementioned QSL-to-normal-metal bilayer that utilizes the QSL material as a spin dissipating subsystem, where spin dissipation via the QSL material results in spin noise generation due to the fluctuation dissipation theorem. The normal metal layer acts as a resistive element, and in the quantum limit, i.e., at very low temperatures, the power spectrum of the asymmetrized spin noise in the metal is entirely quantum in nature [110] and encodes information about the spin sector of the QSL layer. We show that the asymmetrized spin noise generated in this manner must affect, e.g., the ac resistance measured in the normal metal, if the normal metal possesses strong spin-orbit coupling. We then use our theory to examine three QSL models: the S = 1/2 nearest neighbor Heisenberg antiferromagnet on the kagome´ lattice, a model consisting of gapless fermionic spin excitations coupled CHAPTER 5. SPIN NOISE AND QSLS 68

to an emergent U(1) gauge field, and the bare Kitaev honeycomb model in the gapless spin liquid phase. We consider a gapped ground state in our approach to the kagome´ lattice antiferromagnet owing to DMRG studies that indicate the presence of a gap [138], in addition to gapped out flux excitations, and utilize a bosonic parton mean-field theory [7] when characterizing the emergent low energy spin excitations. In so doing, we show that our proposed bilayer system can successfully quantify any spinon gap present. Motivated by the fact that the slave-rotor representation [156] of the Hubbard model on a triangular lattice results in stable QSL mean-field states comprised of a spinon Fermi surface coupled

to a U(1) gauge ﬁeld [8] that may apply to existing candidate materials [e.g., κ-(BEDT-TTF)2-

Cu2(CN)3 and YbMgGaO4], we examine this QSL model theoretically in our proposed bilayer system. We find that a QSL model coupling gapless fermions with a Fermi surface to an emergent U(1) gauge field produces a subdominant Ω4/3 frequency correction to the ac resistance measured across the normal metal that our proposed system can educe. Finally, our treatment of the bare, gapless Kitaev model considers the isotropic point of the quantum spin liquid phase, where all bond interactions are equal in strength, and assumes a flux- free background — a valid assumption for temperatures lower than 1% of the bond strength [157; 158]. Under these considerations we find that when coupled to a normal metal, the presence of a Kitaev QSL will allow characterization of the two-flux gap energy that emerges in spite of the gapless nature of the fermion spectrum. This chapter is structured as follows: Sec. 5.2 presents the theoretical underpinnings of the proposed metal-to-QSL bilayer system, and Sec. 5.3 focuses the discussion on three QSL models that are relevant to the current state of the literature. Within Sec. 5.3, Sec. 5.3.1 considers the Heisenberg kagome´ lattice model with an eye to expanding our understanding of the compound herbertsmithite, Sec. 5.3.2 considers our proposed system utilizing a model comprised of a gapless fermionic QSL with a spinon fermi surface and gapless U(1) gauge fluctuations with the intent of CHAPTER 5. SPIN NOISE AND QSLS 69

understanding certain eﬀectively triangular lattice compounds, and Sec. 5.3.3 relates our proposed system to the Kitaev model at its isotropic point in the QSL phase with the intention of better understanding compounds thought to present a Kitaev QSL phase. We then present physical estimates of the signal strength that should be found in each case in Sec. 5.4, and close with a discussion centered around relating each model and our predictions to real materials thought to possess the relevant QSL phases in Sec. 5.5.

5.2 Theory

Let us consider coupling an insulating spin system (i.e., a quantum magnet) to a strongly spin- orbit coupled heavy-element metal, as depicted in Fig. 5.1. The addition of the spin system gives rise to increased spin dissipation, which, as a result of the fluctuation-dissipation theorem (FDT), contributes additional spin fluctuations inside the metal. If strong spin-orbit interactions are present in the conductor (as in, e.g., Pt, Ta, W [159; 160]), then FDT requires that spin fluctuations across

the interface will give rise to voltage ﬂuctuations in the conductor via the ISHE, i.e., S = S 0 + δS , where S is the total voltage noise in the metal in thermal equilibrium, S 0 is the total noise present in the bare metal, and δS is the portion of the noise that arises due to the presence of the quantum magnet. In the quantum limit — when the sample is cooled to as low a temperature as possible — background thermal noise in the metal is strongly suppressed, and

S (Ω) = 4~ΩR(Ω)θ(Ω) . (5.1)

Here, the Heaviside step function θ(Ω) signiﬁes that only positive frequencies are possible in the quantum limit. This indicates that the metal is able to absorb quasiparticles from an external system, e.g., a detector, but is unable to emit them [110]. The voltage ﬂuctuations present in the CHAPTER 5. SPIN NOISE AND QSLS 70

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metal IAAAV4nicjVhZb9tGEGbSK1WPJG391Beigo30xbDsnG+xI0Nq7kS2E0QyjCW5EhktD++u5EMh0OcCfSgKtC/9Q/0J/TedJamDM5RaAaKo+b75ODs7O1zSSUSg9NbWP1eufvTxJ59+du3z2hdffvX19Rs3vzlS8Ui6/NCNRSzfOkxxEUT8UAda8LeJ5Cx0BH/jDB8Z/M2YSxXE0YG+SPhxyAZR0A9cpsHU+enEPblR39rcyj42PWkUJ3Wr+Lw8ufnd3z0vdkchj7QrmFLdxlaijydM6sAVPK31RoonzB2yAe/CacRCro4nWaypvQ4Wz+7HEr6RtjProseEhUpdhA4wQ6Z9hTFjnGHri2DEz/R5Bpcj0P37x5MgSkaaR24eQH8kbB3bJiG2F0juanEBJ8yVAYzBdn0mmashbbXa+kF73+60X7w+eHR40KmttyTnQ5uJxGcO17UeXNWNw5BF3qTHRGoOAKVlwOGpOWhsH4B5wOAPsnuG73FBHIydJyoQcYSQS0Au6RV840Cs2g9Sc6RIEKfmQOxDlppDkmBAGEBAwXkYeWqQp1VIaK4ejpA1AmOEjQkYkwAZpQlS+jEyK7CqYECyeQH2UWXS+kbdx/LnYHWJ9QysccgHWP0N2F9U2JtmrpoVc9gCe6uYdMkXkQ4gnYoBNJcoaZNhzXDO9sH6NiDqJg2QhRRq2qwSDgdoA8IuCAol3jN0s6ahqjHEZdrdPob8wTKZsex6I134s52Sol708jK6V8GT8Sjy0m4DmIL39S0g2j0ZDHz9IxmTOh0xyefk7px8jGTdkRQXc2ZvUm/kxB4OgDlqzvswV/yAFcfczUeUhWxW+CCIoA9KBrmGVmM8ezBCu8cBzu34YkZEp92dpSoQZq++Dd+d/1CBoO/+H5Ve/TZ878D37gpFqI60e3s6B0jO6G3kchugNleZzxPSArEHy4Kbym3kwW3U72xAaL36vY36/Y36gxVBytUzMA1ztUA/H+eS0OaxlUdaNQOrlPAMrI5p1Vy6OKoiYcsVWTRYKHwBfwWfl3VPZga8WquckEDJf5WYI1lVANXLash1efmtUo4FS/IKqJTOrJvZElwhAvXJRb4GK1Uy/yKanZVKDj9Np/PFoTeZTRaeDZ5xzGwtY+yVVKJiUlGTX1CpZjihmKmEI6HNFpHEknGMypSxTfqxw9lMh4lgQAcUsUKkEnZUMvNXsKfVxD8jGP9K2IvPzNbA/MA44zMEj5Ls9j6FyvcH0x+gSGAHnE1alb+Yc8wUV1GibGsSR6PQgXsYWq2OuQcHwuNVGzzdnKFVN3BXuWaDFjrx+SQ7xzsaPsezc7Ku0gkcY+GZLXMsbEkYzzJG335GkE7Zd9IhfcNRqvBWBErQlRMyq2Yb5TNdYp3TIh0ppGR2hicKJzIn5rmkfOrALyouT9YIv6xgXZIgAxRiQBjvEeM9YTxGjMeEMUaMMU0WYoSEMUSMIWUwnG+wYNIp4pwSxi5i7BLGPmLsE4aPGLS+XyHGK8LYQ4w9wnAQw6FlnoVqNsWwH57sk3Xgt/IqsVsYgMeNaQHlTx5AaHJ4Kpb8Gci9SLhkOpawrTeNYHqF5lKahgV9IJehCprrxByW4PAEC+E0zRH1yHFaWurkPqBaC+G1CNouUOXKSZuguwsoKQO3vaBMfP12kT6COO1yyNS1U7h2COJ06KquaG++oyp4irbBfh+1yj7VSrJH6kxszKT5h1XOywOirTAqE+iN9KJMIN3MN493RRDmFMNHBXhEkKhAyDX95wXyHCNj83wLQ61+HeGgrl6VVo06Uf4sS9six7SFq5J4D4t4DwnyuEBI4/VZgZA+6GdvdLJ85q91SAUVMOkp/l6B0H50ybP3FwsD2iLerwvv1wTZLRCy0Px3BfIOI7ozu3GStaJn2xc6BofJrCSZrKhFJvsF1qsodv1kJvuEYFlSc9C8MsNwNEfNhgrDgT/Dyf1Cz5rCfLy4bPvpbIXWaic36g38qpOeHG1vNnY2t1/drj/c/Tl/DXrN+t76wbplNax71kOrbb20Di3XGli/Wn9Yf655a7+s/bb2e069eqV4dfqtVfqs/fUv1ux6ng== c ⊙

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Figure 5.1: A cartoon of the proposed system: a quantum spin liquid to normal metal bilayer in equilibrium, where fluctuating spin current Is across the interface becomes fluctuating charge current Ic (or fluctuating voltage) in the normal metal via the inverse spin Hall effect and results in measurable modifications to the ac resistance of the metal. metal are thus entirely due to the presence of the QSL candidate material and any background quantum noise. We assume that in the quantum limit the base resistance is essentially constant in the frequency range of interest and known in a given strongly spin-orbit coupled metal, and therefore any quantum background noise linear in frequency that arises can be accounted for and removed, exposing δS (Ω). In the upcoming sections, we provide the technical calculations necessary to extract the correction to the equilibrium voltage noise δS (Ω) — the enhancement to the ac power spectrum of the voltage noise in a strongly spin-orbit coupled metal interfaced with a general quantum magnet possessing no long-ranged order — and then apply our results to some well-known QSL models. Characteristic features in the frequency distribution will then allow for discriminating between the QSL ground states of the various candidate materials via a relatively straightforward near- equilibrium measurements of the ac resistance present in the metal layer via Eq. (5.1). We consider a bilayer system as shown in Fig. 5.1, comprised of a thin normal metal layer (thickness d) affixed atop a QSL material. The interface is set at the xz plane, and the QSL is

treated as a 2d lattice of ﬁxed quantum spins Si. The particular lattice structures of the QSLs come CHAPTER 5. SPIN NOISE AND QSLS 71

into play later in the discussion. We assume that the entire heterostructure is thermalized to a temperature T in order to eliminate the eﬀects of nonequilibrium drives. We assume the QSL and the normal metal are coupled via exchange interaction

X Hc = v0 s(y = 0, ri) Si , (5.2) −J i ·

where v is the volume in the metal per spin of the QSL, is the exchange constant, and r is a 0 J i

vector of the interfacial coordinates at the y = 0 plane specifying the position of Si. The local spin

† density of the metal is given by s(x) = (1/2)ψs(x)τss0 ψs0 (x), where τ is the vector of Pauli matrices and s labels the electron spin quantum number.

The interfacial spin current operator Is is deﬁned by considering the total z polarized spin entering the metal, i.e.,

v0 X + + Is(t) = ( ι)J [s−(y = 0, ri, t)S i (t) s (y = 0, ri, t)S −i (t)] , (5.3) − 2 i −

where ι = √ 1. The spin current noise across the interface can then be computed to lowest non- − trivial order in using, e.g., the real-time Keldysh formalism and the result formally takes the J form [161]

!2 v0 X + h + + i Is(t)Is(0) = J s−(y = 0, ri, t)s (y = 0, r j, 0) S i (t)S −j (0) + S −i (t)S j (0) , (5.4) h i 2 i j h i h i h i where represents correlation functions with respect to the unperturbed Hamiltonian. The h· · · i spectrum of the interfacial spin current noise is then deﬁned via

Z ∞ S (Ω) = dt I (t)I (0) eiΩt . (5.5) s h s s i −∞

The spin correlation function in the metallic sector can be readily computed, so we obtain a CHAPTER 5. SPIN NOISE AND QSLS 72 general expression for the equilibrium spin current noise spectrum at ﬁnite temperature [161],

!2 X Z v0mkF ∞ ν Ω 2 h + + i Ω, = ι ν r r χ − ν + χ− ν , S s( T) 2 J 2 d β~(ν −Ω) sinc (kF i j ) i j ( ) i j ( ) (5.6) 2π ~ e − 1 | − | i j −∞ − where kF is the Fermi wavevector of the metal, and we have introduced the dynamical spin correlation function of the QSL,

Z iνt χ∓±(ν) ι dt S ∓(t)S ±(0) e , (5.7) i j ≡ − h i j i to account for the portion of the noise that arises due to spin ﬂuctuations in the QSL. Finally, for large Fermi wavevectors, i.e., kF ri r j 1 for all i, j, Eq. (5.6) can be approximated in spatially − local terms, and we obtain

!2 Z v0mkF X ∞ ν Ω + + Ω, ι ν χ − ν + χ− ν . S s( T) 2 J 2 d β~(ν −Ω) ii ( ) ii ( ) (5.8) ≈ 2π ~ e − 1 i −∞ −

We emphasize here that the derivation has so far assumed no particular form for the QSL Hamiltonian. Instead, Eq. (5.8) is constructed in order to apply to noise generated in a strongly spin-orbit coupled normal metal due to the proximity of a general quantum magnet (albeit with no long-ranged magnetic order), such that extracting a usable quantity depends only on characterizing

χ±∓ii (ν) for a given QSL model. The ISHE finally converts the z polarized spin current density in the y direction into a charge current density along the x direction (see Fig. 5.1). The conversion ratio between the interfacial spin current fluctuations and the measurable voltage fluctuations may depend non-trivially on various properties of the metal, e.g., its spin Hall angle, geometry, and spin diffusion length. However, as shown in, e.g., Refs. [63; 161], these details may be lumped into a single phenomenological spin-to-voltage noise conversion constant Θ so that the ac voltage fluctuations generated by the CHAPTER 5. SPIN NOISE AND QSLS 73 quantum magnet can be expressed in terms of the spin fluctuations in Eq. (5.8) as

δS (Ω, T) = ΘS s(Ω, T) . (5.9)

Therefore, what we have calculated in Eq. (5.8) is directly related to the ac voltage noise in the metal generated by the presence of the quantum magnet. We may then extract the modiﬁcation to the resistance acquired by the presence of the quantum magnet from this quantity using Eq. (5.1).

5.3 Application to Quantum Spin Liquids

We now consider three types of QSL models in the context of our proposed bilayer system: the antiferromagnetic Heisenberg S = 1/2 kagome´ lattice model (HKLM); a model involving a spinon Fermi surface coupled to an emergent U(1) gauge ﬁeld; and the Kitaev honeycomb model in the gapless spin liquid phase. We ultimately show that the bilayer system we propose will allow the extraction of characteristic frequency dependencies in each case. All three models above are believed to be of relevance for the descriptions of some QSL candidate materials. Detailed discussion of these connections will be presented in Sec. 5.5.

5.3.1 Heisenberg antiferromagnet on the kagome´ lattice

The HKLM is an important prototypical model that supports a QSL ground state, and is believed to be a relevant model for the mineral compound ZnCu3(OH)6Cl2 (herbertsmithite), one of the prime contenders for an experimental realization of a QSL. Recent neutron scattering experiments suggest that the relevant theoretical model for the QSL state observed in this compound consists of the so-called Z2 spin liquids [7; 162; 163]. These QSL states possess two types of excitations, i.e., the S = 1/2 spinon and a gapped vortex, also known as a vison, corresponding to an emergent

Z2 gauge field [118]. It was shown recently that these visons can act as a momentum sink for the CHAPTER 5. SPIN NOISE AND QSLS 74 spinons and can flatten the dynamic spin structure factor [163] in accordance with experimental observations of herbertsmithite. However, for the purposes of the heterostructure we are proposing in this work, we note that only local spin fluctuations are important and visons do not themselves carry spin. Therefore, we expect that visons should not significantly renormalize spin current fluctuations, and that the re-shuffling of the spin quasiparticle spectral weight in momentum space should not drastically affect local quantities. We therefore ignore the effects of the visons in this subsection, follow the calculations performed in Ref. [7] for the HKLM, and apply the result to the observable we are proposing as a probe of the QSL state. We begin with the Hamiltonian for the bare HKLM

J X HHKLM = Si S j , (5.10) −2 i, j · h i where i, j indicates nearest neighbor pairings and J is the exchange coupling. Mean-ﬁeld ap- h i proaches to characterizing the HKLM ground states include bosonic representations [7] of the gapped spinons and fermionic representations, where a gap arises due to pairing in the Hamilto- nian [164]. In what follows, we focus on the bosonic representation of the QSL phase as done in Ref. [7].

† We begin with the standard Schwinger boson representation of the spin operators Si = (1/2)biστσσ0 biσ0 , where b represents a bosonic spinon, and then use the mean-field decoupling Q = (1/2) ε b b , iσ i j h σσ0 iσ jσ0 i where ε is the completely antisymmetric tensor of SU(2) and the field satisfies Q = Q . Sub- σσ0 i j − ji stituting this representation into Eq. (5.10), the mean-field decoupling results in the Hamiltonian

J X X X HHKLM = Qi jεσσ b† b† + h.c. + λ b† biσ , (5.11) 0 iσ jσ0 iσ −2 i, j σσ i,σ h i 0 where h.c. denotes the hermitian conjugate and λ is a Lagrange multiplier that constrains the model to one boson per site. CHAPTER 5. SPIN NOISE AND QSLS 75

eˆ1 eˆ2

eˆ3

Figure 5.2: The kagome´ lattice, with positions on a representative unit cell marked. The eˆi represent the unit vectors of a unit cell, and a is the length of a bond. Arrows between sites correspond to the direction dependent Qi j. Solid arrows around upward facing triangles depict Qi j = Q1, and wire arrows around downward facing triangles depict Qi j = Q2.

Figure 5.2 shows a depiction of the kagome´ lattice, where the arrows between sites in a unit

cell correspond to the direction dependent Qi j, i.e., Q1 and Q2 are the two distinct expectation

values of Qi j. In the ﬁgure, solid arrows correspond to Qi j = Q1 and wire arrows correspond

to Qi j = Q2. The two possible locally stable QSL mean-field solutions [7; 162; 165; 163] occur when Q = Q = Q, corresponding to π flux through the hexagonal plaquette, and Q = Q , 1 2 1 − 2 corresponding to no flux through the plaquette, both with Q , Q R. Here, we will specifically 1 2 ∈ consider the π flux case and utilize mean-field theoretical values put forward in Ref. [163] for our purposes.

P + Upon diagonalizing Eq. (5.11), we can solve for the total spin susceptibility χ(ν) = i[χii−(ν)+

+ χ−ii (ν)], and use it to derive the noise correction. The three basis vectors for the unit cell shown in Fig. 5.2 are a a eˆ = 1, √3 , eˆ = 1, √3 , eˆ = a ( 1, 0) , (5.12) 1 2 2 2 − 3 − where a is the lattice constant. These unit vectors allow us to Fourier transform the Hamiltonian CHAPTER 5. SPIN NOISE AND QSLS 76

into momentum space, where we deﬁne k = eˆ k. Equation (5.11) then reads i i ·    λ †  X  Ck H = Ψ†   Ψ , (5.13) HKLM k   k k Ck λ 

T where Ψ(k) = [b (k), b (k), b (k), b†( k), b†( k), b†( k)] is the vector of particle-hole boson 1 2 3 1 − 2 − 3 − operators for each of the three sites of the unit cell, and the matrix C is a traceless 3 3 matrix k × with components

ik ik c = JQ e 1 + JQ e− 1 = c∗ 12 1 2 − 21 ik ik c = JQ e 2 + JQ e− 2 = c∗ 23 1 2 − 32 ik ik c = JQ e 3 + JQ e− 3 = c∗ . 31 1 2 − 13

Here, Q , Q R are the two distinct expectation values of Q (see Fig. 5.2), and as mentioned 1 2 ∈ i j

previously we explicitly consider the regime Q1 = Q2 = Q. Diagonalizing Eq. (5.13) with a unitary matrix U leads to

X HHKLM = γ†kDγk (5.14) k

with the rotated boson operators γk = U†Ψk. The diagonalized matrix D = Diag (1, 2, 3, 1, 2, 3), where 1 = λ and

p = = λ2 4J2Q2[cos2 (k ) + cos2 (k ) + cos2 (k )] . 2 3 − 1 2 3

P + + To calculate the total spin susceptibility χ(ν) = i[χii−(ν)+χ−ii (ν)], we re-express the sum over P P3 all lattice sites i by reformulating it as i = Nu n=1, where Nu is the number of unit cells and n CHAPTER 5. SPIN NOISE AND QSLS 77

indexes each position within a single unit cell. We then obtain

3 6 2π~Nu X X X k k q q h χ(ν) = ι U ∗ U U ∗U δ(~ν + ξk + ξq ) (k) (q) − N2 nm nm nl¯ nl¯ m l Nm Nl u k,q n=1 m,l=1 i + δ(~ν ξ ξ ) (k) (q) , (5.15) − km − ql Mm Ml where U k is the 6 6 matrix that diagonalizes Eq. (5.11),n ¯ = n+3, and ξ = ( , , , , , ) ml × k 1 2 3 − 1 − 2 − 3 is the vector of energy eigenvalues with the last three negated. Here, we also introduce vectors of distribution functions,

(k) = [n ( ), n ( ), n ( ), n¯ ( ), n¯ ( ), n¯ ( )] N B 1 B 2 B 3 B 1 B 2 B 3 (k) = [n¯ ( ), n¯ ( ), n¯ ( ), n ( ), n ( ), n ( )] , M B 1 B 2 B 3 B 1 B 2 B 3 where 1 n (x) = (5.16) B eβx 1 −

andn ¯ B = 1 + nB. With this, we can express the ac, ﬁnite temperature noise created by the proximate QSL by using Eq. (5.15) in Eq. (5.8),

!2 4π v0mkF X X k k q q S (Ω, T) = J U ∗ U U ∗U s ~N 2π2~ nm nm nl¯ nl¯ u k,q n,m,l " # ξkm ξql ~Ω ξkm + ξql ~Ω − − − m(k) l(q) + − m(k) l(q) . (5.17) × e β(ξkm+ξql+~Ω) 1 N N eβ(ξkm+ξql ~Ω) 1M M − − − −

We are interested in the zero temperature limit of this result. The expression for the zero temperature ac noise power spectrum then becomes

!2 4π v0mkF X X k k q q S (Ω, 0) = J U ∗ U U ∗U (~Ω k q )θ(~Ω k q ) , (5.18) s N ~ 2π2~ nm nm nl¯ nl¯ − m − l − m − l u k,q n,m,l CHAPTER 5. SPIN NOISE AND QSLS 78

00.58.6 6060

50 units)

0.435 . 4040 units) 30 .

0) (arb 20 ,

0.29 10

0.3 ⌦ (

0 00 0.720.5 1.441 2.1.516 0) (arb

, ~⌦/ 0.145 ⌦ ( s 2s = 0.158 S

0 00 0.26 00.52.75 0.78 11.04.5 ~⌦/

Figure 5.3: A numerical plot showing S s(Ω, 0) in the HKLM as a function of ac frequency normalized by the mean-field coupling, λ = 0.695J. The main plot shows the computed sum. The graph is identically zero until a critical frequency of twice the gap energy is achieved, ~Ωc = 2∆s = 0.158λ, at which point it becomes possible to create spinon pairs in the QSL and noise enhancement commences. The inset is the ac susceptibility χ(Ω, 0) in the case of the HKLM, which can be extracted by numerical differentiation of Eq. (5.8), and is in good agreement with previous works, see, e.g., Ref. [3]. where U k is the 6 6 matrix that diagonalizes Eq. (5.11), is the m-th energy eigenvalue of the nm × km diagonalized Hamiltonian, and θ(x) is the Heaviside theta function. Note thatn ¯ = n + 3, the index n = 1, 2, 3, and m, l = 1,..., 6. We now analyze this expression numerically. Figure 5.3 is a plot of Eq. (5.18), from which we find that, in the quantum limit, it is possible to extract an estimate of the spin gap using our proposed bilayer. We use the variable values proffered in Refs. [3; 163] for the HKLM, taking as a reasonable estimate for the spin gap ∆s = 0.055J. In the HKLM, the gap energy can be found by setting k = 0 and finding the minimal eigenvalue [165], p with the outcome ∆ = λ2 12J2Q2. Therefore, rather than computing the values of λ and Q s − variationally, we set λ = 0.695J and tune Q such that ∆s = 0.055J, resulting in Q = 0.2. Note that Q quantifies antiferromagnetic correlations and is restricted to Q 1/ √2, at which point nearest | | ≤ neighbor spins form singlets and the model experiences a phase transition into an ordered phase. Importantly, the HKLM models a highly frustrated antiferromagnetic material, and therefore includes quantum fluctuations. At very low temperatures, quantum fluctuations are required in CHAPTER 5. SPIN NOISE AND QSLS 79

order to drive the interfacial equilibrium noise generated as a result of the mechanism we are considering. Figure 5.3 shows that no noise enhancement is expected until probing frequencies

greater than the critical frequency ~Ωc = 2∆s, at which point production of a pair of spinons can occur in the QSL and an enhancement in the noise begins to manifest. Therefore probing this region of the frequency range for a given material that is adequately modeled by the HKLM should

allow for a direct estimate of the spin gap, ∆s, present in that material.

5.3.2 Spinon Fermi surface coupled to an emergent

U(1) gauge ﬁeld

Some QSL candidate materials, e.g., organic salt compounds [141; 8; 142] and YbMgGaO4 [166; 167; 168], may be described as spinon Fermi seas, where the spin susceptibility is strongly renormalized by an emergent U(1) gauge field. In this subsection, we first compute (using the Keldysh functional integral formalism) the thermal spin current fluctuations arising in the metal due to the presence of fermions in the QSL material, and second how the gauge field renormalization affects those thermal fluctuations in our proposed bilayer system at finite temperature. We then take the T 0 limit at the end and present the zero temperature ac frequency result. → Let us begin with the real-time action for flavors of fermionic spinons coupled to a compact N U(1) gauge field in 2 + 1 dimensions [169; 8; 170]

X Z n 1 2 o S = dtdx c¯σ(t, x)(ι~∂t + µ)cσ(t, x) c¯σ(t, x)[ ι~∇ + a(t, x)] cσ(t, x) , (5.19) σ − 2ms −

where cσ(t, x) is a spinon Grassmann field, σ is the flavor index, ms is the spinon effective mass, µ is the spinon chemical potential, and a(t, x) is the gauge field. The (real-frequency) retarded RPA propagator for the gauge fluctuations in the Coulomb gauge can be obtained by analytically CHAPTER 5. SPIN NOISE AND QSLS 80

continuing the standard result from the imaginary-time formalism [169; 171]

! R qξqξ0 1 1 Dξξ (q, Ω) = δξξ , 0 0 2 2 EFs Ω − − q χdq ι 3 N π~ 3Fsq ! − qξqξ δ 0 dR(Ω) , (5.20) ≡ − ξξ0 − q2 q

where ξ, ξ0 label the Cartesian components, 3Fs = ~kFs/ms is the spinon Fermi velocity, kFs is

1 the spinon Fermi wavevector, EFs is the spinon Fermi energy, and χd = (24~ms)− is the Landau diamagnetic susceptibility of the fermions. Lastly, we define the interaction portion of the Keldysh action by placing Eq. (5.19) on the Keldysh contour and extracting the term dependent on the gauge field to first order,

Z 1 ∞ X X X ξ η ξ η η S int = dt η3 c¯kσ(t )a (t )ck σ(t ) , (5.21) − k+k0 s k k0 0 2~ √A kk σ ξ=x,z η= − −∞ 0 ±

where η = represents the Keldysh time-loop forward and backward branches, 3 = ~k/m , and ± ks s A is the total area of the QSL.

Noise correction due to bare susceptibility

† The spin operators of the QSL are represented using Abrikosov fermions, i.e., Siσ = (1/2)ciστσσ0 ciσ0 , where the constraint of one fermion per site is enforced. Starting from the calculation of the bare bubble, without gauge ﬁeld renormalization, we can ﬁnd the uncorrected noise enhancement gen-

(0) P + (0) erated across the interface by ﬁrst calculating the total susceptibility, χ (ν) = i χii− (ν) + +(0) χ−ii (ν) , given by

X Z (0) ινt + + χ (ν) = 2ι dt e TKc¯i (t−)ci (t−)¯ci (0 )ci (0 ) , (5.22) ↓ ↑ ↑ ↓ − i h i CHAPTER 5. SPIN NOISE AND QSLS 81

where TK is the Keldysh time ordering operator. Equation (5.22) quantiﬁes the portion of the susceptibility that arises due simply to the presence of the fermions in the QSL material, without the additional corrections from emergent gauge photons, and evaluates to

a2m2 ν χ(0)(ν) = ιN s s , (5.23) − π~2 1 e β~ν − − where N is the total number of lattice points and as is the area occupied by each QSL spin. By inserting Eq. (5.23) into Eq. (5.8) and then moving to the zero temperature limit the zeroth order ac noise correction is therefore found to be

!2 N v mk m a 2 S (0)(Ω, 0) = J 0 F s s Ω3 . (5.24) s 3π 2π2~ ~

Gauge ﬁeld correction to the noise

The next non-zero correction to the susceptibility arises due to gauge ﬁeld renormalization and is given by the second order expansion using Eq. (5.21) as the perturbation,

Z (2) ι iνt + + 2 χ∓± (ν) = dt e TKc¯i (t−)ci (t−)¯ci (0 )ci (0 )S . (5.25) ii 2 h ↓↑ ↑↓ ↑↓ ↓↑ inti

Expanding this gives five possible diagrams, only three of which contribute and must therefore be considered; these three are diagrammatically depicted in Fig. 5.4. It is important when calculating gauge invariant quantities to sum over all contributing diagrams, because only then do the divergences cancel exactly. When considering a single gauge propagator, there are three contributing diagrams. Fig- ures 5.4a and 5.4b, where the gauge propagator does not cross the particle-hole bubble, are self- energy corrections to a fermion line, and Fig. 5.4c, where the propagator does cross the bubble, is a vertex correction. Maintaining the gauge invariance of the susceptibility requires that all three CHAPTER 5. SPIN NOISE AND QSLS 82 diagrams are included in the calculation [171; 172]. We begin calculating the susceptibility with the action given in Eq. (5.19), and place it explicitly on the Keldysh contour in the Keldysh basis. We write the components of the action zeroth and first order in the gauge field as

  1    −    R K   1  Z Z   g (t t0) g (t t0) c (t0) X    kσ − kσ −   kσ  S 0 = dt dt0  1 2      , (5.26) c¯kσ(t)c ¯kσ(t)     kσ  A   2   0 g (t t0) c (t0) kσ − kσ         Z   aξ,c (t) aξ,q (t) c1 (t) X  k k0 k k0   k0σ  1 ξ    − −    S 1 = dt 3  1 2      , (5.27) − k+k0 s c¯kσ(t)c ¯kσ(t)     2~ √2A kk     0σξ aξ,q (t) aξ,c (t) c2 (t) k k0 k k0 k0σ − − where 3ks = ~k/ms. We have split the gauge ﬁeld into its classical and quantum components denoted by the c and q superscripts, and split the fermion ﬁelds into their 1 and 2 components in accordance with Ref. [87]. Here, S 1 is the RAK basis equivalent to what we have called S int previously, in Eq. (5.21). Expanding Eq. (5.25), and restricting ourselves to terms with a single gauge propagator, it is

(2) + (2) +(2) possible to write three expressions of the form χn (p, ν) = χn − (p, ν) + χ−n (p, ν), each corresponding to one of the diagrams in Fig. 5.4:

X X X Z Z (2) 1 dΩ dω η1η2 ξ ξ0 χ (p, ν) = η1η2D ( q, ω)3 3 a 4~2A 2 2π 2π ξξ0 − − 2k+2p+qs 2k+2p+qs kq ξξ0 η1η2

h η1 η1η2 η2+ + i g− (Ω + ν)g (Ω + ν + ω)g (Ω + ν)g −(Ω) , (5.28) × k+p k+p+q k+p k

X X X Z Z (2) 1 dΩ dω η1η2 ξ ξ0 χ (p, ν) = η1η2D ( q, ω)3 3 b 4~2A 2 2π 2π ξξ0 − − 2k+2p+qs 2k+2p+qs kq ξξ0 η1η2

h + +η1 η1η2 η2 i g− (Ω + ν)g (Ω)g (Ω + ω)g − (Ω) , (5.29) × k k+p k+p+q k+p CHAPTER 5. SPIN NOISE AND QSLS 83

X X X Z Z (2) 1 dΩ dω η1η2 ξ ξ0 χ (p, ν) = η1η2D ( q, ω)3 3 c 4~2A 2 2π 2π ξξ0 − − 2k+qs 2k+2p+qs kq ξξ0 η1η2

h η η + +η η i g− 1 (Ω + ν)g 1 (Ω + ν + ω)g 2 (Ω + ω)g 2−(Ω) . (5.30) × k+p k+p+q k+q k

Here η , η = represent Keldysh indices for the forward and backward contours, and bare su- 1 2 ± ± perscripts on the Green functions also represent the externally ﬁxed forward and backward Keldysh indices. We now introduce the fermion self-energy in order to represent Eqs. (5.28) and (5.29) in terms of the self-energy. Note at the outset that because the system we consider is in equilibrium, only the retarded spinon self-energy need be explicitly constructed; the advanced self-energy can be derived from the retarded term. The retarded one-loop self-energy can be calculated by considering the second-order correction to the retarded Green function using S 1 as a perturbation

R(2) ι D 1 1 2E g (t, t0) = c (t)¯c (t0)S . (5.31) kσ 2 kσ kσ 1

Extracting the self energy is then a matter of expanding this quantity and enforcing causality, resulting in

X X R 2ι h R K K R i ξ ξ0 Σ (t, t0) = g (t t0)D ( q, t t0) + g (t t0)D ( q, t t0) 3 3 . kσ (4~)2A k+qσ − ξξ0 − − k+qσ − ξξ0 − − 2k+qs 2k+qs q ξξ0 (5.32)

Completing the ξ and ξ0 sums and transforming to frequency space, we have

Z X !2 R ι dω k qˆ h R K K R i Σkσ(Ω) = | × | gk+qσ(Ω + ω)d q( ω) + gk+qσ(Ω + ω)d q( ω) . (5.33) π − − −2A 2 q ms − −

In general we expect d q( ω) to be dominated by small q, due to the form of the propagator. By − − CHAPTER 5. SPIN NOISE AND QSLS 84 selecting the coordinates q = q kˆ + ˆz kˆq we can express the dispersion as k × ⊥

~2q2 ζk+q k µ + ⊥ + ~3Fsq , (5.34) ≈ − 2ms k

2 2 2 with k = ~ k /2ms. From this, we further expect to ﬁnd momentum scaling as q q q , i.e., k ∼ ⊥ ⊥ where q is essentially normal to k, which allows us to replace the momentum dependence of the gauge propagator with q and write k qˆ 2 k2 for momentum near the spinon Fermi surface. ⊥ | × | ≈ Fs Therefore, by explicitly substituting in the spinon Green functions, Eq. (5.33) becomes

2 Z Z Z R ιkFsas dω dq dq Σ Ω ⊥ k kσ( ) 2 ≈ − 2ms 2π 2π 2π  K !   d q ( ω) ζk q ~(Ω + ω)   − ⊥ ⊥ R   − 2πιδ Ω + ω 3Fsq tanh dq ( ω) .  ζk q k  × Ω + ω ⊥ 3Fsq + ιδ − − ~ − ~ − 2kBT ⊥ − − ~ − ~ − k (5.35)

At this point it is clear that the q integral poses no trouble, so we complete it and ﬁnd k Z Z " ! ! # R 3Fsas dω dq 1 ~ω h R A i ~(Ω + ω) R Σ Ω ⊥ ω ω + ω . kσ( ) 2 coth dq ( ) dq ( ) tanh dq ( ) ≈ 2~ 2π 2π −2 2kBT ⊥ − − ⊥ − 2kBT ⊥ − (5.36) Note that from this expression for the self-energy we can see that the k dependence drops out entirely. Evaluating the sums over the Keldysh indices in the self-energy correction terms, Eqs. (5.28) and (5.29), allows us to rewrite the expressions for diagrams a and b in terms of spinon Green’s functions and the self energy:

Z " ! !# 1 1 X dΩ ~Ω ~(Ω + ν) h i χ(2)(p, ν) = ι tanh tanh gR(Ω) gA(Ω) a − 1 e β~ν 2A 2π 2k T − 2k T k − k − − k B B h i gR (Ω + ν)ΣR (Ω + ν)gR (Ω + ν) gA (Ω + ν)ΣA (Ω + ν)gA (Ω + ν) , (5.37) × k+p k+p k+p − k+p k+p k+p CHAPTER 5. SPIN NOISE AND QSLS 85 Z " ! !# 1 1 X dΩ ~Ω ~(Ω + ν) χ(2)(p, ν) = ι tanh tanh b − 1 e β~ν 2A 2π 2k T − 2k T − − k B B h i h i gR (Ω + ν) gA (Ω + ν) gR(Ω)ΣR(Ω)gR(Ω) gA(Ω)ΣA(Ω)gA(Ω) . (5.38) × k+p − k+p k k k − k k k

In the end we must sum together all three diagrams in order to maintain gauge invariance, which

(2) (2) (2) we begin by combining these two terms into χab (p, ν) = χa (p, ν) + χb (p, ν), and use the fact (2) that any integral over only retarded or advanced terms vanishes, to write χab (p, ν) compactly. The outcome is

Z Z Z " ! !# m a 1 dΩ ∞ dζ dθ ~Ω ~(Ω + ν) (2) p s s k k χab ( , ν) = ι β~ν tanh tanh 2~ 1 e− 2π µ 2π 2π 2kBT − 2kBT − −  A h R A i R R h A R i A  gk(Ω) Σk+p(Ω + ν) Σk(Ω) gk+p(Ω + ν) gk(Ω) Σk+p(Ω + ν) Σk(Ω) gk+p(Ω + ν)  − + −  ,  ζk p ζk p  ×  ν + ζk + + ιδ ν + ζk + ιδ  ~ − ~ ~ − ~ − (5.39)

where we have represented the k integral in polar coordinates, with θk the angle between k and p,

and performed a change of variables from k to ζk. Here it is important to point out two things: ﬁrst, the Fermi surface µ is the largest energy scale in the problem, and so we extend the lower bound of the ζ integral to . Second, note that ζ ζ = + ~3 p cos θ ~3 p cos θ , where the k −∞ k+p − k p Fs k ≈ Fs k

last step is due to the expectation that small p dominates, which means the ζk dependence drops

out of the denominator in both terms. Thus there is no ζk dependence in either the denominator

or the self-energies, and so we can complete the ζk integral by simply integrating over the spinon Green functions; this is the reason terms that only have retarded or advanced Green functions in them vanish. The result is

Z Z " ! !# m a 1 dΩ dθk ~Ω ~(Ω + ν) χ(2)(p, ν) = s s tanh tanh ab − 2~ 1 e β~ν 2π 2π 2k T − 2k T − − B B  ΣR Ω + ν ΣA Ω ΣA Ω + ν ΣR Ω   k+p( ) k( ) k+p( ) k( )   − −  . (5.40) × (ν 3 p cos θ + ιδ)2 − (ν 3 p cos θ ιδ)2  − Fs k − Fs k − CHAPTER 5. SPIN NOISE AND QSLS 86

We can now substitute in the self energies represented in Eq. (5.36) to present the ﬁnal step in combining these two diagrams,

2 Z Z Z Z " ! !# (2) kFsas 1 dΩ dω dθk dq ~(Ω + ν) ~Ω χ p, ν = ⊥ ab ( ) 2 β~ν tanh tanh − 2~ 1 e− 2π 2π 2π 2π 2kBT − 2kBT  h − i  ~ω R A ~(Ω+ν+ω) R ~(Ω+ω) A coth 2k T dq ( ω) dq ( ω) tanh 2k T dq ( ω) + tanh 2k T dq ( ω)  B ⊥ − − ⊥ − − B ⊥ − B ⊥ −  2 ×  (ν 3Fs p cos θk + ιδ) − h i  ~ω R A ~(Ω+ν+ω A ~(Ω+ω) R  coth 2k T dq ( ω) dq ( ω) + tanh 2k T dq ( ω) tanh 2k T dq ( ω) + B ⊥ − − ⊥ − B ⊥ − − B ⊥ −  . (5.41) 2  (ν 3Fs p cos θk ιδ)  − −

The third diagram corresponds to a vertex correction. We expect that p k dominates, and recall that the form of the gauge propagator restricts us to small q as well. We express q in components perpendicular and parallel to k as above, and note that, via the same approximation as in Ref. [172], near the Fermi surface we can write

  X ! ~2k2 2 q2 qξqξ0 ξ ξ0 4 Fs q q q  2 δ 3 3  ⊥ + k k ⊥  43 . (5.42) ξξ0 2 2k+qs 2k+2p+qs 2  2 2 2  Fs − − q ≈ − ms q q − q ≈ − ξξ0

Above, the ﬁrst approximation is from recalling that p k and the second is a result of q q2 k ∼ ⊥ q . Therefore, summing over ξ and ξ0 and expanding the Keldysh indices, Eq. (5.30) becomes ⊥

2 Z Z " ! !# 3 1 X dΩ dω ~(Ω + ν) ~Ω χ(2)(p, ν) = F tanh tanh c −2~2A 2 1 e β~ν 2π 2π 2k T − 2k T − − kq B B ( " ! R R A A ~ν h i gk+p(Ω + ν)gk+p+q(Ω + ν + ω)gk+q(Ω + ω)gk(Ω) coth dq ( ω) dq ( ω) × 2kBT ⊥ − − ⊥ − ! !# R ~(Ω + ν + ω) A ~(Ω + ω) dq ( ω) tanh + dq ( ω) tanh − ⊥ − 2kBT ⊥ − 2kBT " ! A A R R ~ν h i + gk+p(Ω + ν)gk+p+q(Ω + ν + ω)gk+q(Ω + ω)gk(Ω) coth dq ( ω) dq ( ω) 2kBT ⊥ − − ⊥ − ! !#) A ~(Ω + ν + ω) R ~(Ω + ω) + dq ( ω) tanh dq ( ω) tanh . (5.43) ⊥ − 2kBT − ⊥ − 2kBT CHAPTER 5. SPIN NOISE AND QSLS 87

Representing the k integral in polar coordinates, where again the angle is with respect to the mo-

mentum p, and then performing a change of variables from the modulus k to ζk, we can complete

the q and ζk integrals. Note that the only quantities that contain these variables now are the spinon k Green’s functions, and those occur only in speciﬁc pairings. We ﬁnd

Z Z dζk dq k gR (Ω + ν)gR (Ω + ν + ω)gA (Ω + ω)gA(Ω) 2π 2π k+p k+p+q k+q k k /v2 = − Fs Fs ~ ν 3Fs p cos θk + ιδ ν 3Fs p cos θk q p sin θk + ιδ − − − ms ⊥

Z Z dζk dq k gA (Ω + ν)gA (Ω + ν + ω)gR (Ω + ω)gR(Ω) 2π 2π k+p k+p+q k+q k k /v2 = − Fs Fs ~ ν 3Fs p cos θk ιδ ν 3Fs p cos θk q p sin θk ιδ − − − − ms ⊥ −

(2) which we can use in χc (p, ν). The result is

2 Z Z Z Z " ! !# kFsas 1 dΩ dω dθk dq ~(Ω + ν) ~Ω χ(2) p, ν = ⊥ c ( ) 2 β~ν tanh tanh 2~ 1 e− 2π 2π 2π 2π 2kBT − 2kBT  − h i coth ~ν dR ( ω) dA ( ω) dR ( ω) tanh ~(Ω+ν+ω) + dA ( ω) tanh ~(Ω+ω)  2kBT q q q 2kBT q 2kBT  ⊥ − − ⊥ − − ⊥ − ⊥ − ×  ~ ν 3Fs p cos θk + ιδ ν 3Fs p cos θk q p sin θk + ιδ ms h − i − − ⊥  coth ~ν dR ( ω) dA ( ω) + dA ( ω) tanh ~(Ω+ν+ω) dR ( ω) tanh ~(Ω+ω)  2kBT q q q 2kBT q 2kBT  + ⊥ − − ⊥ − ⊥ − − ⊥ −  , ~  ν 3Fs p cos θk ιδ ν 3Fs p cos θk q p sin θk ιδ − − − − ms ⊥ − (5.44)

which we must combine with Eq. (5.41) in order to maintain gauge invariance. In order to express the combined term more concisely, we perform the angular integral and CHAPTER 5. SPIN NOISE AND QSLS 88

introduce the function

Z dθk 1 1 I±(q) = q 2π x cos θk ιδ x cos θk sin θk ιδ − ± − − kF ± x = r | | , (5.45) 2 (x ιδ)2 (x ιδ)2 q + 1 ± ± − kF

where x = ν/(3Fs p) and we have substituted q q for notational convenience. Therefore the ⊥ → sum of Eq. (5.41) and Eq. (5.44) becomes

Z Z Z " ! !# k a2 1 1 dΩ dω dq ~(Ω + ν) ~Ω χ(2) p, ν = Fs s ( ) 2 2 β~ν tanh tanh 2~ (3Fs p) 1 e− 2π 2π 2π 2kBT − 2kBT ( −! ~Ω h R A i + + coth dq ( ω) dq ( ω) I (q) + I−(q) I (0) I−(0) × 2kBT − − − − − ! ~(Ω + ν + ω) h R + + A i tanh dq ( ω) I (q) I (0) dq ( ω) I−(q) I−(0) − 2kBT − − − − − ! ) ~(Ω + ω) h R A + + i tanh dq ( ω) I−(q) I−(0) dq ( ω) I (q) I (0) . (5.46) − 2kBT − − − − −

(2) 2 R (2) Recall that we are attempting to obtain the momentum integrated susceptibility: χ (ν) = (2π)− dp χ (p, ν).

Only the angular term I±(q) carries external momentum, however, so the momentum integral can be carried out: Z Z 1 ∞ pdp ∞ dx tan− (q /kFs) ± = ± = ⊥ . 2 I (q) I (q) (5.47) 0 p 0 x − q/kFs

At this point note that the integrand in Eq. (5.46) is even in q, so it is possible to restrict the q P integral to half the domain. If we recall that there is an external sum over all lattice points, i = N, then because the quantum spin liquid is isotropic we have for the ﬁnal result

2 2 Z Z " 1 # 1 Nm a dω ∞ dq h i tan− (q/k ) χ(2)(ν) = s s F(ν, ω) dR( ω) dA( ω) 1 Fs , 1 e β~ν 2(2π)2~4 2π k q − − q − − q/k − − 0 Fs Fs (5.48) a) a) CHAPTER 5. SPIN NOISE AND QSLS 89

b) b) a)

(a) (b) c) b) c)

(c)

Figure 5.4: Three diagrams contributing to the (1/ ) correction to the spin susceptibility gener- c) O N ated by the presence of U(1) gauge ﬁeld ﬂuctuations. with

Z dΩ h i h i F(ν, ω) = tanh ~(Ω+ν) tanh ~Ω 2 coth ~ω tanh ~(Ω+ω+ν) tanh ~(Ω+ω) . 2π 2kBT − 2kBT 2kBT − 2kBT − 2kBT (5.49) These two equations are what we require to progress. Accounting for the required diagrams, the total susceptibility at two-loop order χ(2)(ν) =

P + (2) +(2) i[χii− (ν) + χ−ii (ν)] is given by Eq. (5.48). We can then express the second order ac voltage noise correction as

!2 2 2 Z Z Z v mk m a ν Ω dω ∞ F(ν, ω) δS (2)(Ω, T) = ιN J 0 F s s dν − dz s 2π2~ (2π)2~4 eβ~(ν Ω) 1 2π 1 e β~ν − − 0 − − h R A i 1 1 d q( ω) d q( ω) 1 z− tan− z . (5.50) × − − − − − −

We now move into the zero temperature limit, which allows completion of each of the integrals. We expand the q integral to lowest order in ω for ω Ω , where Ω is the spinon Fermi Fs Fs CHAPTER 5. SPIN NOISE AND QSLS 90 frequency. Thus, the ac noise correction due to gauge ﬂuctuations ﬁnally becomes

!4/3 0.075 Ω δS (2)(Ω, 0) = S (0)(Ω, 0) . (5.51) s Ω s N Fs

We can therefore see from the zeroth and second order calculations that the total ac voltage fluctuations generated across the normal metal sample when interfaced with a QSL material possessing an emergent U(1) gauge field will have a Ω3 lowest order frequency dependence modified by a Ω4/3 subdominant component. The total ac noise correction generated is given by the sum of the two terms, such that the power spectrum of the enhancement becomes

 !4/3 (0)  0.075 Ω  S s(Ω, 0) = S (Ω, 0) 1 +  , (5.52) s  Ω  N Fs when considering the regime Ω Ω . Fs That a noise enhancement accrues due to the presence of an emergent gauge field is physically understandable as the collective fermionic spin excitations present, i.e., the spinons, acquiring a higher scattering probability due to the added presence of gauge fluctuations. Additional scattering must result in additional noise generation. This enhancement should be extricable in the quantum limit using Eq. (5.1): first plotting δR(Ω)/Ω2 and then subtracting off the intercept reveals the bare Ω4/3 correction term characteristic of the gauge field renormalization calculated in Eq. (5.52).

5.3.3 The Kitaev Honeycomb Model

The second example of a Z2 QSL is the Kitaev model on the honeycomb lattice (see Fig. 5.5), where exchange frustration arising due to the inability to simultaneously satisfy all Kitaev interactions along neighboring bonds can drive the system into a QSL phase. The Kitaev spin liquid is exactly solvable, and we select this example for consideration in our proposed system for that reason, in addition to the fact that there are potential material candidates available currently [123; 136]. CHAPTER 5. SPIN NOISE AND QSLS 91

Furthermore, we restrict our investigation to the gapless phase, where the Kitaev exchange cou-

plings are equal. However, while fermionic excitations are gapless, the emergent Z2 gauge ﬁeld is not, and this has the peculiar eﬀect of generating an apparent gap in the fermionic sector [4; 173]. The Kitaev model therefore provides something of a synthesis of the last two sections, as it is both

a Z2 and gapless theory. We give a short overview of the approach to solving the Kitaev model following Refs. [9; 174], and then apply the solution to extracting an observable out of our proposed heterostructure. The Kitaev model on the honeycomb lattice is given by [9]

X H = K S γS γ , (5.53) K − γ i j γ, i, j γ h i where γ = x, y, z represents the different nearest neighbor bond directions (see Fig. 5.5) at each { } lattice point with interaction strength Kγ. In understanding the gapless spin liquid phase of the Ki- taev model, parton mean-field theories have been proposed [9; 175] that characterize the emergent excitations as Dirac fermions [174] arising in tandem with an emergent gapped flux. This can be seen by first representing the spin operators in terms of four Majorana fermions, i.e., S γ = ι f c with f , f = 2δ δ , c , c = 2δ , and f , c = 0, which then gives S γS γ = i iγ i { iγ i0γ0 } ii0 γγ0 { i i0 } ii0 { iγ i0 } i j P ιuˆ c c , whereu ˆ = ι f f is the bond operator. Noting that the bond operators commute − i j i j i j γ iγ jγ

with each other and with any bilinear operator containing ci, it is possible to replace them with their eigenvalues 1. One thus obtains S γS γ = ιc c , and H becomes bilinear in the Majorana ± i j ± i j K fermions. Second, we note that the so-called ﬂux operator on a plaquette Wp = f1x f2y f3z f4x f5y f6z, where the subscript p labels the plaquette number, commutes with the Hamiltonian and is therefore an integral of motion. When a plaquette has an even number of bonds, as Fig. 5.5 makes clear is the case for the honeycomb lattice, its eigenvalues are 1. Finally, we note that the spin representation ± in terms of four Majorana fermions enlarges the Hilbert space from two to four, and must therefore be constrained in order to recover the physical Hilbert space. This constraint is enforced via a CHAPTER 5. SPIN NOISE AND QSLS 92

n2 n1

Figure 5.5: The Kitaev honeycomb lattice, where the two sub-lattices are shown by the purple (sub-lattice A) and orange (sub-lattice B) dots. Representative x-, y-, and z-links are depicted. The ﬂux operator on plaquette p, Wp, is shown, n1 and n2 are the primitive lattice vectors, and a is the lattice spacing. Also depicted is a representative unit cell enclosed in the dashed box.

projection operator Pi = (1/2)(1 + fix fiy fizci) for each site i, which requires that the initial spin algebra be conserved. Effectively, what has been done is to reduce the initial Hamiltonian to a noninteracting Dirac fermion hopping Hamiltonian in a static Z2 gauge field, where the choice of values foru ˆ i j amounts to fixing a gauge, and the gauge invariant quantities are the plaquette operators Wp. A theorem by Lieb [176] guarantees that if the number of sites per plaquette is 2 mod 4, then the ground state is in the flux-free sector, i.e., where it is possible to setu ˆ i j = 1. The ground state is therefore described by the free Majorana hopping Hamiltonian,

X HK = ιK cic j , (5.54) i, j h i where K = K = K K, and the subscripts i and j are now labeling A and B sites respectively. x y z ≡ Equation (5.54) can be expressed in terms of complex fermions using a standard procedure [4]. We take r as a unit cell coordinate, comprised of an A site and a B site as shown in Fig. 5.5, and

combine the two Majorana species into a single complex fermion species through br = (cAr +

ιcBr)/2. Putting the system on a torus with Nu unit cells, and changing to reciprocal space with CHAPTER 5. SPIN NOISE AND QSLS 93

1/2 P ιkr br = Nu− k e bk, Eq. (5.54) becomes

    X  ε ι∆   b   k k   k  HK = b† b k     , (5.55) k −     k     ι∆k εk b† k − − −

ιkn1 ιkn2 where εk = Re(sk), ∆k = Im(sk), sk = K(1 + e + e ), and the primitive vectors n1 and n2 are P defined in Fig. 5.5. Thus the ground state energy is E = s [9; 4]. 0 − k | k| The isotropic Kitaev spin liquid offers a unique opportunity in that only on-site and nearest neighbor spin correlations contribute [177]. Therefore we use Eq. (5.6) in calculating the noise and include both on-site and nearest neighbor contributions under the assumption that the envelope sinc2(k r r ) does not noticeably vary over the lattice constant a. We then find that the total F| i − j| P + + susceptibility χ(ν) = i j[χi j−(ν) + χ−i j (ν)] becomes

Z χ(ν) = 8ιN dt S x (t)S x (0) + S x (t)S x (0) eινt , (5.56) − u h A0 A0 i h A0 B0 i where we invoke the translation invariance of the isotropic spin liquid to select the r = 0 unit cell. The above two dynamical spin correlation functions can be written as [177]

x x ιHK t/~ ι(HK +Vx)t/~ S A0(t)S A0(0) = e cA0e− cA0 K (5.57) x x ιH t/~ ι(H +V )t/~ S (t)S (0) = ι e K c e− K x c , A0 B0 − A0 B0 K where the subscript K for the expectation values indicates they are taken with respect to the ground state of H , and the bond potential reads V = 2ιKc c with i and j, again, representing the A and K x − i j its adjacent x-bond connected B site, respectively [177; 4]. The form of the correlators in Eq. (5.57) can be understood by taking into consideration the

γ effect of operator S i on an eigenstate. In addition to adding a single Majorana fermion ci at site i, the spin operator also adds one π flux each to the two plaquettes sharing the γ-bond emanating from i. The bond potential Vx represents the insertion of this flux pair, and Eq. (5.57) calculates CHAPTER 5. SPIN NOISE AND QSLS 94

the dynamic rearrangement of the Majorana fermions at time t following the sudden appearance of the fluxes at t = 0. Incidentally, Baskaran et al. showed that this quench problem is equivalent to an exactly solvable x-ray edge problem [177]. It was later shown by Knolle et al. [4] that due to a vanishing Majorana density of states in the Kitaev model, slowly switching on the bond potential in the infinite past and then switching it off in the infinite future — what may be called the adiabatic approximation — replicates the quench dynamics in the low energy limit. Thus, as we desire a probe of the low energy density of states, we perform the calculation under this approximation, meaning we re-write Eqs. (5.57) so that the

correlation functions are taken with respect to the Hamiltonian Hx = HK +Vx, i.e., the Hamiltonian with a ﬂux pair present, or equivalently one ﬂipped x-bond. As a result, we evaluate

x x ιE0t/~ ιHxt/~ S A0(t)S A0(0) e cA0e− cA0 x ≈ (5.58) x x ιE t/~ ιH t/~ S (t)S (0) ιe 0 c e− x c , A0 B0 ≈ − A0 B0 x

where the subscript x on the correlators explicitly indicates that they are now taken with respect

to the ground state of Hx. Time-evolving the Majorana operators in Eqs. (5.58) in the Heisenberg

ιH t/~ ιH t/~ picture ci(t) = e x cie− x , we write

x x ι∆F t/~ S A0(t)S A0(0) = e− cA0(t)cA0(0) x (5.59) x x ι∆ t/~ S (t)S (0) = ιe− F c (t)c (0) , A0 B0 − A0 B0 x where the two-ﬂux gap energy ∆ 0.26K is the energy required to insert the ﬂuxes [9]. F ≈

Replacing the Majorana operators in the correlation functions with the complex fermions bk allows us to re-write the total susceptibility Eq. (5.56) as

Z ι(ν ∆ /~)t χ(ν) = 16ιN dte − F b (t)b†(0) . (5.60) − u 0 0 x CHAPTER 5. SPIN NOISE AND QSLS 95

The last correlator can be obtained by solving a Dyson equation. We recall that the unperturbed

Hamiltonian for the b fermions is HK as given by Eq. (5.55) where the components written out explicitly are ε = K(1 + cos q n + cos q n ), ∆ = ιK(sin q n + sin q n ), and n , n are the q · 1 · 2 q − · 1 · 2 1 2 real space lattice vectors as shown in Fig. 5.5. The local potential across the x-bond of the r = 0 unit cell is given by u u X u Vx = ub0†b0 = b†pbq , (5.61) − 2 Nu pq − 2

where u = 4K and N is the number of unit cells in the lattice. Then the full Hamiltonian − u

including the local potential is, as mentioned above, Hx = HK + Vx, for which we want to ﬁnd the Green functions.

We take the equation of motion approach, so we require the commutation relations for bk,

u X [Hx, bk] = 2εkbk + 2∆kb† k bk . (5.62) − − − Nu k

We now examine the retarded matrix Green functions in the Nambu basis, which we express as GR (t, 0) = θ(t)[G> (t, 0) G< (t, 0)] (note that capitalized G Green functions will be reserved for pq pq − pq matrices, and lower-case g for components). The greater-than matrix Green function is

       bp(t)b† q(0) bp(t)bq(0)  >  h − i h i  G (t, 0) = ι   , (5.63) pq −      b† p(t)b† q(0) b† p(t)bq(0)  h − − i h − i and the lesser-than Green function matrix can be found by anti-commuting the fermions in each component. Using Eq. (5.62), we then ﬁnd

2 u X ι∂ GR (t, 0) = ιδ(t)δ + H (p)GR (t, 0) + σ GR (t, 0) , (5.64) t pq pq ~ 0 pq ~ z pq N p

where H0(p) is the coeﬃcient matrix of HK, and σz is the z-component Pauli matrix. Fourier CHAPTER 5. SPIN NOISE AND QSLS 96

transforming to frequency

u X GR (ω) = GR,0δ + GR,0(ω)σ GR (ω) , (5.65) pq p pq ~ p z kq Nu k

where the unperturbed Green function matrix for the bk fermions reads

    ω + 2 ε + ιδ 2 ∆  1  ~ p ~ p  R,0   G p (ω) =   , (5.66) p(ω)   N  2 2   ∆∗ ω εp + ιδ ~ p − ~

and (ω) = (ω + ιδ)2 (2/~)2 s 2. When summing over the momentum index, we can see that Np − | p| R,0 the oﬀ diagonal elements of G p (ω) vanish because ∆p is an odd function of p, and the Brillouin zone is symmetric about the origin. This is the reason there are no anomalous Green functions

of the form b†(t)b†(t0) — or the complex conjugate — as mentioned previously. Thus we have h 0 0 i P R,0 G (ω) = Diag gR (ω), gR (ω) , where the elements are p p { 11 22 }

2 X ω εp + ιδ gR (ω) = ± ~ . (5.67) 11,22 (ω) p Np

We then sum over p in Eq. (5.65) in order to rewrite the full Green function in terms of unperturbed Green functions as

    1 X  X  − R  u  R,0   R,0 G (ω) = 1  G (ω) σz G (ω) . (5.68) pq  ~  p   q p − Nu p

Finally, by substituting this expression back into Eq. (5.65) we ﬁnd

     gR (ω)   11   u R ω 0  X  1 ~N g11( )  GR (ω) =  − u  . (5.69) pq   pq  gR (ω)   22   0 1+ u gR (ω)  ~Nu 22 CHAPTER 5. SPIN NOISE AND QSLS 97

P The correlation function we require is b (t)b†(0) = (1/N ) b (t)b†(0) , which in fre- h 0 0 ix u pqh p q ix quency space is ιg> (ω). In equilibrium at the quantum limit, G>(ω) = θ(ω)[GR(ω) GA(ω)], and 11 − therefore we have   gR (ω) gA (ω) >  11 11  g11(ω) = θ(ω)   . 1 u gR (ω) − 1 u gA (ω) − ~Nu 11 − ~Nu 11 Utilizing the result, we produce (at zero temperature)

R A 16~θ(~ν ∆F)[g (~ν ∆F) g (~ν ∆F)] χ(ν) = − 11 − − 11 − , (5.70) h 4K R i h 4K A i 1 + g (~ν ∆F) 1 + g (~ν ∆F) Nu 11 − Nu 11 −

R where the local retarded Green function g11(ω) is shown in Eq. (5.67), and is explicitly

X R x + 2εk A g (x) = = g ∗(x) . (5.71) 11 (x + ι0+)2 4 s 2 11 k − | k|

The ac noise correction generated in the metal by proximity to a gapless Kitaev spin liquid can now be calculated by utilizing Eq. (5.70) in Eq. (5.8) in the quantum limit. We ﬁnd

!2 Z 32ι v mk S (Ω, 0) = J 0 F dε(~Ω ε)θ(~Ω ε) s ~ 2π2~ − − R A θ(ε ∆F)[g (ε ∆F) g (ε ∆F)] − 11 − − 11 − , (5.72) h 4K R i h 4K A i × 1 + g (ε ∆F) 1 + g (ε ∆F) Nu 11 − Nu 11 −

which can be measured electrically as ac resistance via Eq. (5.1). Figure 5.6 depicts our numerical evaluation of Eq. (5.72) in the main plot, with the critical

frequency Ωc = ∆F/~ marked. The ﬁgure displays our primary result for the gapless Kitaev spin liquid compound, namely, that even in the ostensibly gapless case the ac noise correction generated

in the metal is suppressed until frequencies greater than the two-ﬂux gap, ∆F. Equivalently, via the

FDT, measurement of the ac resistance enhancement in the metal near the critical frequency Ωc should expose the two-ﬂux gap when measured utilizing the proposed heterostructure.

That a completely suppressed region occurs seems odd given that the fermion spectrum sk is CHAPTER 5. SPIN NOISE AND QSLS 98

0.120.12

3 0.1 3 units) .

units) 2 0.08 2 . 0.08

0) (arb 11

0.06 , ⌦ ( 0 0) (arb . 0.04 0 1 2 3 4 5 6 7 , 0 04 0 3 6

⌦ ~⌦/K (

s 0.02 F = 0.26 S

0 00 0.1 0.2 0.3 0.4 00.5.5 0.6 0.7 0.8 0.9 1 ~⌦/K

Figure 5.6: A numerical plot showing S s(Ω, 0) for the Kitaev model as a function of ac frequency normalized by the isotropic interaction strength, K. The main plot shows the computed sum. The graph is identically zero until a critical frequency equal to the gap energy is achieved, ~Ω = ∆ c F ≈ 0.26K, at which point it becomes possible to inject flux pairs into the QSL and noise enhancement commences. Here, N = 4 104 has been used. The inset is the ac susceptibility χ(Ω, 0) in the u × case of the Kitaev model, which can be extracted via numerical differentiation of Eq. (5.8), and is in good agreement with previous works [4]. gapless, however while inserting a fermion into the lattice requires no extra energy input, it also requires the addition of a flux in each of the adjacent plquettes. These fluxes are gapped, and so an indirect gap appears even for the fermions. This outcome has been noted previously [4; 173], however the proposed bilayer system offers a method for quantifying the flux gap experimentally.

5.4 Physical Estimates

In this section we estimate the expected signal strength of the voltage noise arising across the heavy metal layer of our proposed bilayer system as a result of the proximity of a QSL state characterized by each of the three models we examine, noting that Eq. (5.1) connects the voltage noise and the ac resistance. The variable Θ in Eq. (6.12) is a conversion parameter connecting the charge and spin sectors via the ISHE. We estimate the strength of Θ by ﬁrst assuming, given Pt with a spin diﬀusion length CHAPTER 5. SPIN NOISE AND QSLS 99

λ = 2 nm and thickness d = 7 nm, a spin current density proﬁle of

sinh [(d x)/λ] j (x, t) = NI (t) − , (5.73) s s sinh [d/λ]

where N is the number of QSL spins per unit area on the interface and I(t) is the spin current penetrating the interface. We model the spin to charge current conversion via the ISHE, so that

we have jc(x, t) = θSH (2e/~) js(x, t), where θSH is the spin Hall angle. Integrating over the cross- sectional area of the metal and multiplying by the resistivity ρ reveals the total voltage generated across the metal: ! ! 2eλ d V (t) = θ NlρI (t) tanh , (5.74) c SH d~ s 2λ

where l is the length of the metal layer. Then we can re-express Eq. (6.12) using voltage noise, δS = V (t)V (0) , by writing V h c c i

" !#2 2e λ d δS (Ω, T) = θ Nρl tanh S (Ω, T) . (5.75) V SH ~ d 2λ s

8 1 In Pt θ 0.1, ρ 10− Ωm, the lattice spacing a = 4 Å k− , and the electron eﬀective mass SH ≈ ≈ ∼ F

m = 13me [178]. The prefactor here characterizes Θ. In each QSL case, we assume an exchange coupling strength of /k 1 K, set the unitless J B ∼ integral component of S 0.01, and work out N using the lattice spacing of the QSL spin sites. In s ∼ the case of herbertsmithite, sample sizes can have linear dimensions of 1 cm [179], and the lattice spacing is on the order of 10 Å, so N 1020 spins/m2. Given that J/k 200 K [133] and using ∼ B ∼ 18 2 Eq. (5.18) in Eq. (5.75), our proposed bilayer system should ﬁnd δS 10− V s when probing V ∼ herbertsmithite. For the organic triangular lattice compounds, sample sizes can have linear dimensions of 1

1 mm [128], and nearest neighbor spin distances on the order of a 10 Å k− [128], so that ∼ ∼ Fs N 1020 spins/m2. We approximate the spinon velocity in the QSL using that of 1D spin chains, ∼ CHAPTER 5. SPIN NOISE AND QSLS 100

v πJa/2~, as in Ref. [128], resulting in a Fermi frequency of Ω 1014 Hz, and J/k Fs ≈ Fs ≈ B ∼ (0) 18 2 250 K [120] in these compounds. As a result, we find that δS 10− V s, and the sub-leading V ∼ (2) 20 2 correction due to the gauge field δS 10− V s. V ∼ Lastly, we represent Kitaev compounds using α-RuCl , with a lattice constant of a 6 Å, 3 ≈ sample sizes of linear dimension 1 cm, and coupling strength K/k 80 K [136]. Thus, N 1019 B ∼ ∼ 2 18 2 spins/m . Using Eq. (5.72) in Eq. (5.75) results in δS 10− V s. Voltage fluctuations of these V ∼ magnitudes have been measured in, e.g., Ref. [63].

5.5 Discussion and Conclusions

Equation (5.8) is a general relationship that quantifies the noise enhancement present in a normal metal adjacent to an insulating magnet lacking long range magnetic order as a result of the coupling to that magnet, i.e., as a result of spin fluctuations across the interface. We assume that in the quantum limit the base ac resistance of a particular strongly spin-orbit coupled metal layer is known and constant, and as a result that background quantum noise can be accounted for and removed. This exposes the portion of the voltage noise power spectrum that quantifies the presence of the affixed quantum magnet, or more specifically the QSL candidate material. So, by measuring the ac frequency dependence of the spin fluctuations in the metal layer, it should be possible to compare the various QSL candidate materials in order to examine gapped and gapless models, and the low energy spin excitations of these exotic phases of matter in general. Now we will connect the models we have examined to real materials.

The compound herbertsmithite, ZnCu3(OH)6Cl2, evinces no long range magnetic order down to temperatures of at least 50 mK, which is four orders of magnitude lower than its exchange coupling J/k 200 K [133], suggesting the presence of a QSL phase. Additionally, the Cu2+ B ∼ ions form a perfect kagome´ lattice, and therefore herbertsmithite is modeled, to a ﬁrst approximation, by the HKLM. Experimental treatments of herbertsmithite include NMR [121; 122], neutron CHAPTER 5. SPIN NOISE AND QSLS 101

scattering [134; 135], and susceptibility [124] studies, most of which indicate that ground state excitations should be gapless. However, while some numerical DMRG studies agree [140], others

show that the ground state of herbertsmithite should be a gapped spin liquid with Z2 topological order [138], and the more recent NMR measurements indicate a gapped ground state as well [122]. Additional complications arise due to the fact that the low energy spectrum appears to be dominated by impurity spins [180] that occur as a result of Cu2+ ions replacing some non-magnetic Zn2+ in the transition metal sites between kagome´ layers [181]. We have shown that our proposed heterostructure can address some of these issues. For instance, the presence of a gap energy will be indicated by a lack of frequency scaling in the voltage noise power spectrum measured in the heavy metal layer in the quantum limit up to a critical frequency of ~Ωc = 2∆s. Technical issues notwithstanding, herbertsmithite remains one of the most promising QSL candidate material on offer, and our proposed technique adds a tool for further investigation. The primary organic salt candidate QSL material that has been modeled as a spinon fermi sea coupled to an emergent U(1) gauge field is κ-(BEDT-TTF)2-Cu2(CN)3, henceforth κ-ET [8; 141]. It is an antiferromagnetic weak Mott insulator, where structural dimers possessing a single spin-1/2 degree of freedom arise and form an approximately triangular lattice that becomes geometrically frustrated when cooled. The primary experimental evidence of QSL like behavior comes from thermal and NMR measurements; in the specific heat versus temperature, for example, a large linear term is observed [127; 125; 126], and NMR measurements show a lack of long range magnetic ordering down to temperatures of 32 mK, approximately 4 orders of magnitude lower than the exchange coupling of J/k 250 K [120]. The necessity of working at very low temperatures B ∼ when probing the low energy density of states of κ-ET and similar compounds has caused some doubt as to the reliability of specific heat measurements, however, due to the difficulty of properly accounting for nuclear contributions. Nevertheless, other triangular lattice compounds exist, and the recent discoveries of possible QSL phases in YbMgGaO4 [182] and NaYbO2 [183], neither of which order down to temperatures of at least 70 mK, provide new opportunities to study previ- CHAPTER 5. SPIN NOISE AND QSLS 102

ously unknown QSL candidate materials — opportunities the proposed heterostructure is intended to capitalize upon by providing a clear prediction for the frequency scaling of the modiﬁcation to the ac resistance enhancement in the metal. We have shown that in its QSL state, a compound that can be modeled as a spinon fermi sea coupled to a U(1) gauge ﬁeld should evince a correction to the ac resistance measured across a coupled heavy metal that goes as Ω4/3, which would therefore shed light on the low energy density of states directly. The heterostructure proposed here would avoid the issue of contamination by signals due to nuclear spins entirely, and provide a powerful tool for probing the low energy excitations of more recently discovered candidate QSL materials as well.

Finally, we mention the iridate honeycomb materials [184] and α-RuCl3 [185] in the context of Kitaev compounds accessible to our bilayer technique. Initially, iridates like α-Na2IrO3 and

4+ α-Li2IrO3 were realized to have Ir ions, with eﬀective j = 1/2, arranged in a honeycomb lattice. While these particular compounds were found to order at low temperatures [186; 187; 188], this class of materials does exhibit bond-directional Kitaev interactions [189]. Thus the discovery of a

variant that does not magnetically order, H3LiIr2O6, is interesting [123]; in particular, it would be an excellent test of this material to see if the measured resistance enhancement in a coupled heavy metal ﬁlm in fact behaves as we predict should be the case for Kitaev spin liquid compounds.

A second material, α-RuCl3, is also known to evince similar structural properties to the iridates,

3+ with magnetic Ru ions forming a honeycomb lattice. As with most iridates, α-RuCl3 orders at low temperatures [190], though interesting Kitaev spin liquid-like behavior occurs before order-

ing [191; 136]. However, while α-RuCl3 exhibits Kitaev-like physics prior to freezing, questions have arisen as to whether or not Kitaev physics are actually responsible for that appearance, rather than more conventional physics from the perspective of magnon-like excitations [192]. We have shown that the proposed bilayer could perhaps address this controversy directly, as the presence of

Kitaev spin liquid physics in the α-RuCl3 layer should result in characteristic suppression of the ac resistance enhancement in the coupled metal, particularly suppression below any ﬂux gap present. CHAPTER 5. SPIN NOISE AND QSLS 103

We therefore believe that our proposed QSL-to-metal bilayer could be helpful to the search for a material exhibiting a true Kitaev spin liquid state. In conclusion, we have proposed a heterostructure composed of a QSL candidate material over- laid with a strongly spin-orbit coupled heavy metal film as a viable probe of QSL ground states that can perhaps alleviate some of the currently outstanding controversies. Our proposal marries concepts from spintronics and quantum magnetism in order to use the well-understood physics of equilibrium noise in a new context, namely the search for, and categorization of, QSL ground states in various materials. The theory advanced here indicates that an equilibrium measurement, i.e., a measurement taken within the linear response regime, of the ac resistance in the normal metal layer will provide information about the low energy density of states of the QSL material under observation due to a connection via the FDT and Eq. (5.1) to the noise power spectrum we have calculated. In particular, determining the presence or absence of a gap, or probing a gapless QSL as in the U(1) gauge fluctuations case will be readily accessible to this method. For quantifying a gap energy beyond the reach of modern electronic spectral analyzers, quantum absorption measurements utilizing the relaxation rate of a two level system coupled to the metal layer, as in Ref. [193], may prove advantageous. Thus, the system we propose will have wide ranging applica- bility to probing insulating quantum magnets in general, and should provide an interesting method of examining and categorizing QSL candidate materials in particular. We show that the proposed bilayer should be able to extract any gap energy present in the HKLM, that fermions coupled to an emergent U(1) gauge field should see a sub-dominant frequency dependent correction arising in the ac resistance that goes as Ω4/3, and that the bare Kitaev model in the gapless spin liquid phase should nevertheless evince gapped behavior extricable through ac resistance measurements across the coupled metal layer. We have restricted our analysis in this chapter to bulk conversion effects in the metal layer, so in future works it could be interesting to consider interfacial effects that may affect conversion; in this vein examining different metal compounds like CuBi [194] with low intrinsic spin-orbit CHAPTER 5. SPIN NOISE AND QSLS 104 coupling could be illuminating. Additionally, it would be interesting to expand this analysis to include extensions to the QSL models such as Dzyaloshinskii-Moriya interactions, or correction to the Kitaev model [192], and especially to include disorder in the analysis, as disorder appears to be highly influential when attempting to discriminate between QSL and non-QSL ground states. A closer examination of the interface will be the work of the next chapter. Chapter 6

Paramagneto conductance and bilayer interfaces

In this chapter we examine a metal-to-quantum paramagnet bilayer system and develop a microscopic theory of the interface in the low temperature regime. In order to focus exclusively on interfacial physics, the metal is modeled as a 2d electron gas with spin-inert, short range disorder. With low dimensional unconventional magnetic insulators, e.g., quantum spin liquids (QSL), in mind, we model the spin system as a 2d array of fixed spins. Spin fluctuations therefore present at the interface mediate electron-electron interactions with the result that the spin dynamics of the paramagnet govern the types of corrections seen in the metallic conductivity. The presence of an interface breaks inversion symmetry and causes interfacial Rashba spin-orbit coupling to arise which we also include in the calculation. We perform a consistent perturbative calculation to first non-zero order in the spin excitation-electron coupling strength, and extract corrections to the conductance that arise with and without spin-orbit coupling. Ultimately, we find that — contra the case of screened Coulomb interactions — no universal modification arises from the exchange terms; rather, the conductivity modifications that arise depend on the type of paramagnet utilized in the bilayer. We apply the theory developed to the case of a gapless QSL possessing a spinon

105 CHAPTER 6. PARAMAGNETO CONDUCTANCE 106

Fermi surface and show that without spin-orbit coupling a metallic, i.e., enhancing, correction to the conductivity occurs. Introducing spin-orbit coupling results in an insulating correction which dominates the metallic correction when temperatures enter the millikelvin range.

6.1 Introduction

When a non-magnetic conductor (N) is interfaced with a magnetic insulator, spin transfer across the interface can cause corrections to the resistance in the metal [195]. An important example is spin Hall magnetoresistance (SMR), in which exchange interaction at the interface between con- duction electron spins and the magnetization in the insulator leads to a characteristic modulation of the electrical conductivity as the magnetic orientation is varied [32; 31; 36]. SMR serves as a powerful probe not only of the spin-sink and spin-mixing conductances, which govern the spin opacity across N-to-magnet interfaces [32; 33; 34; 35; 39; 44; 45; 43], but also of the magnet’s bulk properties [37; 38; 39; 40; 41; 42]. Several recent experiments extended the SMR technique to bilayer systems coupling metals to paramagnetic insulators (PIs) and reported SMR-like signals under external magnetic fields [46; 47; 48]. However, even in the absence of the field, the PI acts as a bath of spin fluctuations from the point of view of the metal, and these spin fluctuations can have profound effects on the conductivity of the adjacent metal. The effects of spin fluctuations on the adjacent metallic conductivity may become important in the presence of metallic disorder and at low temperatures. Transport measurements in N-PI bilayers performed below approximately 50K indeed show evidence for weak anti-localization [196; 197; 42; 48], which is signaling the inevitable presence of disorder in the metal. Altshuler and Aronov showed that screened Coulomb interactions in a 2d disordered Fermi liquid can lead to strong singularities near the Fermi surface — a ln T-correction to the metallic conductivity at low temperatures now known as the Altshuler-Aronov (AA) corrections [198; 199]. Even in the absence of Coulomb interactions, AA corrections can incite important modifications to metallic conductiv- CHAPTER 6. PARAMAGNETO CONDUCTANCE 107

ity. In a 2D disordered Fermi liquid near a ferromagnetic quantum critical point, where electron interactions are mediated by critical spin fluctuations, an insulating AA correction to the metallic conductivity has been predicted [200]. N-PI bilayers offer an ideal realm in which to study the AA correction arising due to spin fluctuations: unlike the monolayer situation, a multilayer offers tunability, where fluctuation strength can be modified via, e.g., the insertion of an insulating spacer layer between the N and the PI. The array of available PIs also affords the possibility of utilizing systems hosting many types of spin quasiparticles, from magnons to more exotic spin excitations, e.g., spinons. Lastly, breaking inversion symmetry across the interface gives rise to Rashba-like spin-orbit coupling and offers the opportunity to examine AA corrections arising due to spin-orbit coupling in addition to spin fluctuations. In this chapter, we examine theoretically a N-PI bilayer and perform a microscopic calculation of the electrical conductivity across the metal film in the presence of the coupled spin system. We consider a negligibly spin-orbit coupled metal thin film (e.g., Cu, Al) with spin inert, short range impurities, and the cases both with and without Rashba-like spin-orbit coupling arising due to structural inversion symmetry breaking at the interface [201]. The calculation proceeds in the diffusive regime, where kBTτ 1 with τ the elastic electron scattering time, and we develop a ~ general theory for PIs to first non-zero order in the electron-spin excitation coupling. We find that the modifications to the dc conductivity depend on the spin structure of the affixed PI, and apply the general model to the case of a gapless quantum spin liquid (QSL) with a spinon Fermi surface. Without weak spin-orbit coupling, a metallic correction that goes as ln T arises due to the presence of the affixed QSL; with the addition of interfacially generated spin-orbout coupling,

1/2 the correction becomes insulating and goes as T − . The corrections therefore compete, and a crossover temperature in the millikelvin range emerges. Our results for gapless QSLs indicate that AA-type modiﬁcations to the metallic conductivity arising due to the presence of a coupled PI can act as a probe of PI materials — in particular those hosting gapless QSL ground states. The chapter proceeds as follows: we begin in Sec. 6.2 by introducing the model of the bilayer. CHAPTER 6. PARAMAGNETO CONDUCTANCE 108

z metal thin ﬁlm y x ⇥ jc J insulating magnet Figure 6.1: A cartoon of the proposed bilayer system. An insulating magnet, e.g., a quantum spin liquid, is coupled to a weakly spin-orbit coupled metal thin ﬁlm. The interface breaks inversion symmetry causing Rashba spin orbit coupling in the metal layer. As a result, the current in the x direction, jc, carries signatures from both the interfacial spin orbit coupling and proximity to the insulating magnet.

In Sec. 6.3 we will introduce the linear response formalism used in what comes subsequently, and link to the Kubo formalism through the retarded response function. Section 6.4 introduces the diagrams that contribute to the conductivity in the presence of disorder and electron-electron interactions mediated by spin fluctuations, and Sec. 6.5 explicates the disorder averaging procedure and where AA-type disorder physics enters the calculation. In Sec. 6.6 we depict how cancelation of the diamagnetic terms emerges, and present the final result for the ac conductivity in Sec. 6.7. Finally, we discuss our findings and conclude in Sec. 6.8.

6.2 The Model

Let us consider coupling an insulating spin system — in particular a low dimensional quantum magnet — to a weakly spin-orbit coupled, disordered metallic thin film, i.e., effectively 2d, as depicted in Fig. 6.1. We orient the bilayer in the x-y plane, such that inversion symmetry is broken in the z direction by the addition of the metallic thin film. Breaking inversion symmetry across the system gives rise to interfacial Rashba spin-orbit coupling in the metallic layer, which acts independently of the spin neutral random disorder present. Thus, we consider the metal as a non-interacting 2d electron gas (2DEG) with random spin-neutral disorder and interfacial Rashba spin-orbit coupling. CHAPTER 6. PARAMAGNETO CONDUCTANCE 109

The quantum magnet layer is modeled as a 2d array of ﬁxed spins, where each spin interacts with the electron density in the metal via exchange coupling. The exchange coupling is weak, such that the magnetic layer remains essentially undisturbed, and we calculate corrections to the metallic conductivity perturbatively utilizing the weak coupling as our perturbative quantity. With the quantum magnet present, we will therefore calculate the ﬁrst non-zero correction to the conductivity across the metallic layer, i.e., we will compute the correction to the conductivity of a non-interacting 2DEG with spin interactions, spin-neutral disorder, and in the presence of interfacially driven Rashba spin-orbit coupling.

The Hamiltonian of the disordered metallic thin ﬁlm layer is given by H = H0 + HI + HSO, where the subscripts indicate the metallic Hamiltonian, the impurities, and the spin-orbit coupling respectively. That is, we have

X Z ~2∇2 ! H0 = dr ψσ† (r) µ ψσ(r) (6.1) σ − 2m − X Z HI = dr u(r)ψσ† (r)ψσ(r) (6.2) σ ι~2λ X Z H = dr ψ† (r) [∇ (zˆ τ)] ψσ (r) , (6.3) SO − m σ · × σσ0 0 σσ0 where µ is the Fermi surface, ψ (ψ†) is the real space fermion annihilation (creation) operator in P the metal, u(r) = u δ(r r ) is a delta impurity potential for i = 1 ... N impurities, λ is the i − i I spin-orbit coupling strength, m is the eﬀective electron mass in the metal, and τ is the vector of Pauli matrices. The paramagnetic current operator derived from this Hamiltonian is

X p ιe~ h i eλ~ j (r) = ψ† (r) ∇ψσ (r) ∇ψ† (r) ψσ (r) δσσ + [zˆ τ] ψ† (r)ψσ (r) , (6.4) −2m σ 0 − σ 0 0 m × σσ0 σ 0 σσ0 where e < 0 is the electron charge. We assume the metal is exchange coupled to a 2d QSL, and that the interface is held in the x-y CHAPTER 6. PARAMAGNETO CONDUCTANCE 110

plane. The coupling is given by

X Hc = ν0 s(ri) Si , (6.5) −J i · where is the coupling strength, ν is the 2d volume of spin density in the metal that couples to J 0 P s r = / ψ† r τ ψ r a spin in the QSL, the electron spin density in the metal is ( i) (1 2) σσ0 σ( i) σσ0 σ0 ( i),

and Si is the spin operator of a spin at spin site i in the QSL. We utilize Hc as our perturbing Hamiltonian, with as our perturbative quantity, such that the current response function is given J by ι ι R ηη0 p η p η ~ dt00 Hc(t00) Π (r, r0, t t0) = T j (r, t ) j (r0, t0 0 )e− cK , (6.6) ab − −~ K a b where a, b = x, y, z, the superscripts η, η0 = indicate forward (+) and backward (-) branches of ± the Keldysh contour, cK denotes the Keldysh contour, and TK is the Keldysh time ordering operator.

R + + The retarded response function is therefore formed by calculating Π (t) = θ(t) [Π− (t) Π −(t)], − with θ(t) the Heaviside step function.

6.3 Linear response and the Kubo formalism

We wish to determine the response of the metallic layer of this proposed bilayer system to a spatially homogeneous, external electromagnetic field in the presence of an affixed QSL material. We can account for the presence of the electromagnetic field in the metal by using the gauge covariant gradient, i.e., ∇ ∇ ie A(t), where A(t) is the homogeneous, time dependent vector potential, ± → ± − ~ and we select the gauge where the scalar potential vanishes. To first order in the vector potential, the coupling to the electromagnetic environment emerges from Eq. (6.1) and is given by

Z p H0(t) = dr j (r) A(t) . (6.7) − · CHAPTER 6. PARAMAGNETO CONDUCTANCE 111

Using the gauge covariant derivative in Eq. (6.4), we also ﬁnd the gauge invariant, physical charge current operator: 2 X p e j(r, t) = j (r, t) A(t)ψσ† (r)ψσ(r) . (6.8) − m σ

2 The second term is the diamagnetic component of the charge current, jd(r, t) e A(t)ρ(r), where ≡ − m P ρ(r) = σ ψσ† (r)ψσ(r) is the density. Under the assumption that the electric ﬁeld is applied exclusively in the x direction, we write j(r, t) j (r, t) and A(t) A (t). Direct application of the Kubo formula using Eqs. (6.6) and (6.8) ≡ x ≡ x gives the expression for the expectation value of the current along the x direction:

Z Z D d E R j(r, t) = j (r, t) dr0 dt0 Π (r, r0, t t0)A(t0) . (6.9) h i − xx −

These thermal averages are taken with respect to Eqs. (6.1)-(6.3). Impurity averaging restores translational invariance, and so we eventually use the momentum representation. Then the paramagnetic current operator in the x direction is given by

X " # p ~ ~λ y j = e (2q + k ) δσσ + τ σσ a† aq+kσ (6.10) k 2m x x 0 m 0 qσ 0 qσσ0

where a (a†) is the Fourier transformed fermion annihilation (creation) operator. Additionally, the presence of interface driven spin-orbit coupling in the metal gives rise to electron propagators with non-trivial spin structure. The retarded metallic Green function is therefore given as

   ξk ~λ ιθ  1 ω ~ + ιδ ι ke−  GR(ω) =  − m  , (6.11) k 2   (ω ξk + ιδ)2 ~λ k  ι ~λ keιθ ω ξk + ιδ − ~ − m − m − ~

where ξ = ~2k2/2m µ, θ is the angle associated with the momentum k, v = ~k /m is the k − F F Fermi velocity. Note that we have retained the full momentum dependency here, though we will eventually approximate the spin-orbit derived oﬀ-diagonal terms at the Fermi surface. Finally, the CHAPTER 6. PARAMAGNETO CONDUCTANCE 112

a) t1 t2 d) t t1 t2 0 t 0

t2 b) e) t1 t 0 t 0

t2 t1

c) f ) t1 t 0 t1 t 0

t2 t2

Figure 6.2: The six possible diagrams, derived from Eq. (6.13). Solid lines depict fermion propagators, and dashed lines spin propagators. Diagrams a) and b) are the self-energy corrections, diagram c) is the vertex correction, d) is the barbell diagram, and e) and f) are the Hartree diagrams. ac conductivity in the x direction, calculated in the linear response regime, is given as

2 ΠR (k, Ω) + e ρ(Ω = 0) σ(Ω) = lim xx m h i . (6.12) k 0 ιΩ → −

6.4 Contributing diagrams

The ﬁrst non-zero correction to the response function, Eq. (6.6), is the correction second order in the interfacial coupling, . This is due to the fact that the QSL layer lacks long range magnetic J CHAPTER 6. PARAMAGNETO CONDUCTANCE 113

order, such that S = 0. Thus Eq. (6.6) gives us h i

ηη ι D p η p η E Π 0 (r, r0, t) = T j (r, t ) j (r0, 0 0 ) xx −~ K 2 2 X X Z Z ν0 D p η p η α β E αβ J dt dt T j (r, t ) j (r0, 0 0 )s (r , t )s (r , t ) χ (t , t ) ~3 1 2 K i 1 j 2 i j 1 2 − 2 αβ i j cK cK (6.13)

Πηη0(0) + δΠηη0 ≡ xx xx

D E where we have introduced the spin susceptibility of the QSL layer, χαβ(t , t ) = ι T S α(t )S β(t ) , i j 1 2 − K i 1 j 2 and α, β = x, y, z are spin orientations. The spin susceptibility is general, and this calculation therefore applies to any quantum magnet lacking long range magnetic order. We can apply Wick’s theorem to the fermion ﬁelds due to the fact that the Hamiltonian is Gaussian in fermion operators, and we ﬁnd six unique diagrams thereby, as depicted in Fig. 6.2. We will consider each of these diagrams in what follows, starting with the exchange and vertex correction diagrams.

6.4.1 The self-energy terms

The self-energy diagrams with times and real-space coordinates listed are presented in Fig. 6.3.

η1η2 D η1 η2 E After disorder averaging, Green functions of the form G (t t ) = ι T a (t )a†(t ) , where pq 1 − 2 − K p 1 q 2 η , η = are Keldysh indices, become diagonal in momentum space. However, prior to disorder 1 2 ± averaging, they are not. So, we will present the intermediary steps in real space until disorder averaging to regain translational symmetry at which point we transform to momentum space. Ad- ditionally, we set the external momentum to zero throughout, as per Eq. (6.12). The ﬁrst diagram CHAPTER 6. PARAMAGNETO CONDUCTANCE 114

10 t1 2 t2 1 3 ri rj + t r r0 0 t r r0 0 rj ri 40 20 t2 30 t1 4

Figure 6.3: The two self-energy correction diagrams with energy ﬂow around the bubble depicted. Solid lines are fermion propagators, and dashed lines are spin propagators. The real space coordinates are listed, for ease of calculation. Numbers associated with each fermion line correspond to those listed in the Green functions in the main text.

in Fig. 6.3 results in

2ν2e2 X X Z Z ηη0,(a) 0 αβ η1 η2 ηη1,σ00σ1 α η1η2,σ2σ3 β δΠ (r, r0, t) = J dt dt χ (t t )Grr (t t )τ Gr r (t t )τ xx ~3 1 2 i j 1 2 i 1 σ1σ2 i j 1 2 σ3σ4 4 αβ i j cK cK − − −

η2η0,σ4σ r0 η0η,σ0σ000 r G (t2)γσσ G ( t)γσ σ , (6.14) × r j r0 0 r0 r − 000 00 where we have assumed sums over all internal Keldysh and spin indices. The γ variables represent real-space current vertices that we leave general until the transformation back to momentum space at which point we will furnish explicit expressions. The second diagram in Fig. 6.3 must be considered in concert with the ﬁrst, and so we present it here as well. It has a similar form to the ﬁrst and is explicitly given by

2ν2e2 X X Z Z ηη0,(b) 0 αβ η1 η2 ηη0,σ00σ r0 η0η,σ0σ1 α δΠ (r, r0, t) = J dt1 dt2 χ (t t )G (t)γσσ G ( t1)τσ σ xx ~3 i j 1 2 rr0 0 r0 ri 1 2 4 αβ i j cK cK − −

η1η2,σ2σ3 β η2η,σ4σ000 r Gr r (t1 t2)τ Gr r (t2 t)γ , (6.15) × i j − σ3σ4 j − σ000σ00 where again internal Keldysh and spin indices are summed over. Note that in both Eqs. (6.14) and (6.15) the ordering of the elements is consequential due to the spin structure. Combining the diagrams as shown pictorially in Fig. 6.3 produces the correction due to the self- CHAPTER 6. PARAMAGNETO CONDUCTANCE 115 energy. This requires expansion of the Keldysh indices and transformation to frequency space. In order to calculate the correction to the conductivity we form the retarded response function for each diagram. By performing these operations, we ﬁnd that the retarded response function derived from the ﬁrst self-energy diagram becomes

2 2 2 X X Z Z R,(a) ν0e dω d r r α β δΠ (r, r0, Ω) = J γ γ 0 τ τ xx ~3 π π σ000σ00 σσ0 σ1σ2 σ3σ4 − 4 αβ i j 2 2 ( R R R A αβR b() f (ω) f (ω Ω) Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () × − − − −

R R R A αβA b() f (ω) f (ω Ω) Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − − − R R R A αβR + f ( ω) f (ω) f (ω Ω) Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − − − R A R A αβR f ( ω) f (ω) f (ω Ω) Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − − − − R R R R αβR + b() f (ω Ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − A A A A αβR b() f (ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − R R R R αβR + f ( ω) f (ω Ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − − R A R R αβR f ( ω) f (ω Ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − − − R R R R αβA b() f (ω Ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − − A A A A αβA + b() f (ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − A A A A αβA + f ( ω) f (ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () − − − ) A R A A αβA f ( ω) f (ω)Gω(1)Gω (2)Gω(3)Gω Ω(4)χi j () , (6.16) − − − − where we have combined the spin and position indices into the numeric arguments within each Green function to reduce notational clutter to the extent possible — the numbers coincide with those depicted in Fig 6.3 for the (a) diagram. The functions f (ω) and b() are Fermi-Dirac and Bose-Einstein distributions respectively. Additionally, this result has been arranged with cancelation of the diamagnetic terms in mind: terms with well behaved dc limits are grouped, and terms CHAPTER 6. PARAMAGNETO CONDUCTANCE 116 divergent in the Ω 0 limit come after. This will be explored further in upcoming sections. → The second self-energy diagram is denoted the (b) diagram, where the self-energy appears on the bottom leg of the fermion bubble. This has a similar form to Eq. (6.16) and is also grouped to highlight well-behaved terms. The result is given by

2 2 2 X X Z Z R,(b) ν0e dω d r r α β δΠ (r, r0, Ω) = J γ γ 0 τ τ xx ~3 π π σ000σ00 σσ0 σ1σ2 σ3σ4 − 4 αβ i j 2 2 ( R A A A αβR b() f (ω) f (ω Ω) Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () × − − − − − −

R A A A αβA b() f (ω) f (ω Ω) Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − − − R A R A αβA + f ( ω + Ω) f (ω) f (ω Ω) Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − − − R A A A αβA f ( ω Ω) f (ω) f (ω Ω) Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − − − − − R R R R αβR + b() f (ω Ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − A A A A αβR b() f (ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − R R R R αβR + f ( ω + Ω) f ( ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − − R R A R αβR f ( ω + Ω) f (ω Ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − − − R R R R αβA b() f (ω Ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − − A A A A αβA + b() f (ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − A A A A αβA + f ( ω + Ω) f (ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () − − − − − ) A A R A αβA f ( ω + Ω) f (ω)Gω(10)Gω Ω(20)Gω Ω(30)Gω Ω(40)χi j () . (6.17) − − − − − −

Primed numerals represent spin and position indices around the (b) diagram, as depicted in right- most diagram of Fig. 6.3. Green functions in both Eq. (6.16) and (6.17) are given in the same order as in Eqs. (6.14) and (6.15). We additionally make use of the identity

f (ω) (b() + f ( ω)) b() f (ω ) = 0 (6.18) − − − CHAPTER 6. PARAMAGNETO CONDUCTANCE 117

t1 1 2 ri

t r r0 0

rj 4 3 t2 Figure 6.4: The diagram for the vertex correction with energy ﬂow and real-space positions noted. Solid lines are fermion propagators, and the dashed line represents a spin propagator. Numbers associated with each fermion line correspond to those listed in the Green functions in the main text.

in arriving at the ﬁnal expressions for the self-energy diagrams.

6.4.2 The vertex correction terms

The next diagram we treat is the vertex correction, a depiction of which is presented in Fig. 6.4. The explicit expression for the vertex correction prior to expanding the Keldysh contours is

2ν2e2 X X Z Z ηη0,(c) 0 αβ η1 η2 ηη1,σ00σ1 α η1η0,σ2σ r0 δΠ (r, r0, t) = J dt1 dt2 χ (t t )Grri (t t1)τσ σ G (t)γσσ xx ~3 i j 1 2 1 2 ri r0 0 4 αβ i j cK cK − −

η0η2,σ0σ3 β η2η,σ4σ000 r G ( t2)τσ σ Gr j r (t2 t)γσ σ , (6.19) × r0 r j − 3 4 − 000 00 where, again, we note the spin structure dictates the ordering of the elements. As in the case of the self-energy, we proceed by ﬁrst expanding the Keldysh contours and Fourier transforming the result to frequency space, and then forming the retarded current response function. This gives the CHAPTER 6. PARAMAGNETO CONDUCTANCE 118 result

2 2 2 X X Z Z R,(c) ν0e dω d r r α β δΠ (r, r0, Ω) = J γ γ 0 τ τ xx ~3 π π σ000σ00 σσ0 σ1σ2 σ3σ4 − 4 αβ i j 2 2 ( R R A A αβR b() f (ω) f (ω Ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () × − − − − − −

R R A R αβR f (ω Ω) f ( ω + Ω) f ( ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − − − R R A A αβR + f ( ω Ω) f (ω) f (ω Ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − − R A A A αβR f ( ω) f (ω) f (ω Ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − − R R A A αβA b() f (ω) f (ω Ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − A R A A αβA + f (ω) f ( ω + Ω) f ( ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − R R A A αβA f ( ω + Ω) f (ω) f (ω Ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − − R R R A αβA + f ( ω + Ω) f (ω) f (ω Ω) Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − R R R R αβR + b() f (ω Ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − A A A A αβR b() f (ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − R R R R αβR + f (ω Ω) f ( ω Ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − R A A R αβR f (ω Ω) f ( ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − − A A A A αβA + b() f (ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − R R R R αβA b() f (ω Ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − − A A A A αβA + f (ω) f ( ω)Gω(1)Gω (2)Gω Ω(3)Gω Ω(4)χi j () − − − − − A R R A αβA f (ω) f ( ω + Ω)Gω+Ω(1)Gω +Ω(2)Gω Ω(3)Gω Ω(4)χi j () , (6.20) − − − − − − where the spatial and spin indices have been wrapped into the numerical Green function arguments as in the self-energy case, which align with those found in Fig. 6.4. Deriving these expressions requires use of the identity presented in Eq. (6.18), as in the previous subsection. This expression CHAPTER 6. PARAMAGNETO CONDUCTANCE 119

1 3 ✏ + ⌦ ! + ⌦ 5 t1 t 0 t ⌦ 2 ✏ ! 2 4 Figure 6.5: The barbell diagram with energy ﬂow and real space positions noted. Solid lines are fermion propagators, and the dashed line represents a spin propagator. Numbers associated with each fermion line correspond to those listed in the Green function arguments in the main text.

has been arranged mirroring the self-energy terms so that the ﬁrst eight terms have a well-deﬁned dc limit and the last eight do not. We will show that the combination of Eqs. (6.16), (6.17), and (6.20), together with the exchange diamagnetic term, conspire to produce a well-behaved dc limit.

6.4.3 The barbell terms

The barbell diagram is notably simpler than the previous two. We have depicted it visually here in Fig. 6.5, and the formal expression prior to expanding the Keldysh contours is

2ν2e2 X X Z Z ηη0,(d) 0 αβ η1 η2 ηη1,σ000σ1 α η1η,σ2σ00 r δΠ (r, r0, t) = J dt1 dt2 χ (t t )Grr (t t1)τ Gr r (t1 t)γ xx ~3 i j 1 2 i σ1σ2 i σ00σ000 − 4 αβ i j cK cK − − −

η2η0,σ4σ0 r0 η0η2,σσ3 β G (t2)γσ σG ( t2)τσ σ . (6.21) × 0 r0 r j − 3 4

Notice here that compared to Eqs (6.14) and (6.19) an extra negative sign arises due to the presence of a second fermion bubble. After expanding the Keldysh contours and transforming to frequency CHAPTER 6. PARAMAGNETO CONDUCTANCE 120

space, the retarded response function is

2 2 2 X X Z Z R,(d) ν0e dω d r r α β αβR δΠ (r, r0, Ω) = J γ γ 0 τ τ χ (Ω) xx ~3 π π σ00σ000 σ0σ σ1σ2 σ3σ4 i j − 4 αβ i j 2 2 ( h i h i f (ω + Ω) f ( + Ω) GR (4) GA (4) GA(2) GR (2) GA (2) GA(1) × +Ω − +Ω ω+Ω − ω+Ω ω h i h i + f (ω) f ( + Ω) GR (1) GA (1) GA(2)GR (3) GR (4) GA(4) +Ω − +Ω ω+Ω ω − ω h i h i + f (ω + Ω) f ()GA (1) GR(2) GA(2) GR (3) GA (3) GA(4) +Ω − ω+Ω − ω+Ω ω h i h i ) + f (ω) f ()GR (1) GR(2) GA(2) GR (3) GR (4) GA(4) . (6.22) +Ω − ω+Ω ω − ω

Once more, all spin and position indices are subsumed into the numerical argument of the Green functions, and the numbers correspond to those in Fig. 6.5.

6.4.4 The Hartree diagrams

The remaining two diagrams in Fig. 6.2 are the Hartree corrections, shown as (e) and ( f ). We again have two fermion bubbles, as in the barbell correction, only now instead of breaking the bubble apart a tadpole emits from either the top or the bottom fermion line. Figure 6.6 depicts the diagrams in question. The explicit expression for the ﬁrst diagram is

2ν2e2 X X Z Z ηη0,(e) 0 αβ η1 η2 ηη1,σ000σ1 α η1η0,σ2σ0 r0 δΠ (r, r0, t) = J dt1 dt2 χ (t t )Grri (t t1)τσ σ G (t1)γσ σ xx ~3 i j 1 2 1 2 ri r0 0 − 4 αβ i j cK cK − − η0η,σσ00 r <,σ4σ3 β G ( t)γσ σ Gr r (0)τσ σ , (6.23) × r0 r − 00 000 j j 3 4 CHAPTER 6. PARAMAGNETO CONDUCTANCE 121

4 10 ! + ⌦ ✏ t2 rj t 0 5 0 ! ! t1 ri 1 2 + 20 t r 30 ! 1 i ! 50 0 t 0 t2 rj ! ⌦ ✏ 3 40 Figure 6.6: The Hartree diagrams with energy ﬂow and real space positions noted. Solid lines are fermion propagators, and the dashed line represents a spin propagator. Numbers associated with each fermion line correspond to those listed in the Green function arguments in the main text.

where here note that the closed fermion loop brings in twice the lesser-than Green function. The second diagram, with the tadpole on the bottom leg of the diagram, is given by

2ν2e2 X X Z Z ηη0,( f ) 0 αβ η1 η2 ηη0,σ000σ0 r0 η0η1,σσ1 α δΠ (r, r0, t) = J dt1 dt2 χ (t t )G (t)γσ σG ( t1)τσ σ xx ~3 i j 1 2 rr0 0 r0 ri 1 2 − 4 αβ i j cK cK − − η1η,σ2σ00 r <,σ4σ3 β Gr r (t1 t)γ G (0)τ . (6.24) × i − σ00σ000 r j r j σ3σ4

By expanding the Keldysh contours in Eqs. (6.23) and (6.24), transforming to frequency space, and forming the retarded response function for the Hartree diagrams we ﬁnd

2 2 2 X X Z Z R,(e) ν0e dω d r r α β δΠ (r, r0, Ω) = J γ γ 0 τ τ xx ~3 π π σ00σ000 σ0σ σ1σ2 σ3σ4 4 αβ i j 2 2 ( R R A < αβR f (ω) f (ω Ω) Gω(1)Gω(2)Gω Ω(3)G (4)χi j (0) × − − −

R R R < αβR + f (ω Ω)Gω(1)Gω(2)Gω Ω(3)G (4)χi j (0) − − ) A A A < αβR f (ω)Gω(1)Gω(2)Gω Ω(3)G (4)χi j (0) , (6.25) − − for the Hartree diagram with the tadpole on the top branch of the fermion bubble. The second CHAPTER 6. PARAMAGNETO CONDUCTANCE 122

diagram, with the tadpole emerging from the bottom gives

2 2 2 X X Z Z R,( f ) ν0e dω d r r α β δΠ (r, r0, Ω) = J γ γ 0 τ τ xx ~3 π π σ00σ000 σ0σ σ1σ2 σ3σ4 4 αβ i j 2 2

R A A < αβR + f (ω + Ω) f (ω) G (10)G (20)G (30)G (40)χ (0) − ω+Ω ω ω i j R R R < αβR + f (ω)Gω+Ω(10)Gω(20)Gω(30)G (40)χi j (0) ) A A A < αβR f (ω + Ω)G (10)G (20)G (30)G (40)χ (0) . (6.26) − ω+Ω ω ω i j

These two diagrams have been written so as to highlight the terms that do not have a well-behaved dc limit, mirroring the self-energy and vertex correction terms displayed above. We will ﬁnd that Eqs. (6.25) and (6.26) combine with the Hartree correction to the diamagnetic term to produce a well-behaved dc limit.

6.5 Disorder averaging

P The disorder we consider consists of delta potential impurities in the metallic layer, u(r) = u δ(r i −

ri), at locations ri. We remain within the first Born approximation, and consider only non-crossing disorder lines. We treat the metallic Green functions, the current vertices, and quantum interference effects between Green function lines by the insertion of impurity ladders connecting them. In this section, we will present each of the required expressions used to disorder average our result. Disorder averaging begins by the replacement of bare Green functions with disorder averaged Green functions. The disorder averaged Green functions are given by solving the standard Dyson equation [202] to find " # 1 h i 1 1 − (ω) = G (ω) − Σ (ω) , Gp p − ~ p

where, due to the spin structure of the Green functions, the superscript 1 indicates a matrix −

inversion, G for the retarded version is given in Eq. (6.11), and Σp(ω) is the self-energy comprised CHAPTER 6. PARAMAGNETO CONDUCTANCE 123

p p p0 p0 p00 = + + + p00 p0 p ··· p0 p

p0 p = +

p0 p

Figure 6.7: A pictorial depiction of the current vertex correction. The ﬁlled in dot is the corrected current vertex, and the empty dot is the uncorrected vertex. Impurity scattering events are denoted by stars and dashed lines. An impurity ladder is inserted in the top line, and then resummed into the Dyson equation as shown in the second line.

of irreducible diagrams and solved within the ﬁrst Born approximation. Then the self-energy gives

nu2 X Σ (ω) nu + G (ω) , p ~ k ≈ V k

with V the 2d interfacial area, and the ﬁnal result is

   ξp ι ιθ  ω + ιλvFe−  R 1  ~ 2τ  p(ω) =  −  , (6.27) ξ 2  ξ  G ω p + ι λ2v2  ιλv eιθ ω p + ι  − ~ 2τ − F − F − ~ 2τ

1 2 2 where (2τ)− = πnu N0/~ is the scattering lifetime, n is the impurity density, N0 = m/2π~ is the 2d density of states of the metal, and we have approximated the spin-orbit coupling terms at the Fermi surface. This is the disorder averaged retarded Green function for a 2d metal with Rashba spin-orbit coupling and spin-inert delta impurities. An important thing to notice at this point is that after disorder averaging translational invariance is restored and a disorder averaged Green function depends only on a single momentum. It is necessary to use corrected current vertices wherever possible in order to remain within the conserving approximation and therefore maintain current conservation in the calculation. The corrected current vertices are depicted in Fig. 6.7, where after an impurity ladder insertion the CHAPTER 6. PARAMAGNETO CONDUCTANCE 124

Dyson equation for the corrected vertex is formed. Setting the external momentum to zero, as required by Eq. (6.12), the uncorrected vertex can be found by diﬀerentiating the inverse retarded (or advanced) Green function such that

    d h i 1 ~  p ιλ 0 R −  −  γp = G p(ω) =   , (6.28) −dp m ιλ p  where p is the momentum ﬂowing through the vertex. Using the representation in Fig. 6.5 to produce the necessary integral equation, we ﬁnd that the corrected current vertex is given by

2 X 0 nu γp = γ + p (ω)γp p (ω) , (6.29) p V~2 G 0 0 G 0 p0 where the single impurity event on the second line in Fig. 6.7 results in a factor of nu2, and is the G disorder averaged Green function given above, each of which can be either retarded or advanced. Immediately it is clear that the kernel of the integral equation vanishes when the Green functions are not of opposite causality, due to the necessity to have poles on either side of the real axis. We also note that the external momentum p falls out of the summation. Solving Eq. (6.29) results in removal of the anomalous component of the current operator [203] such that we ﬁnd

~p γσσ0 = δ . (6.30) p m σσ0

This is the corrected current vertex we use in calculating the conductivity. We account for quantum interference eﬀects between Green function lines via the insertion of impurity ladders connecting them, following past works in disordered electron systems [204; 205]. An impurity ladder is constructed pictorially in Fig. 6.8. The ladder function is a four-tensor in the spin-sector due to the spin structure of the Green functions. The Feynman rule for a single impurity

(0)σ1σ2 1 scattering event is Lσ4σ3 = (2π~N0τ)− δσ1σ2 δσ3σ4 , and so we construct the Dyson equation for L CHAPTER 6. PARAMAGNETO CONDUCTANCE 125

! ✏ ! ✏ ! ✏ ! ✏ ! ✏ ! ✏ 1 2 1 2 1 2 p0 q p q p0 q p00 q p q p0 q p q + + = p0 p p0 p00 p p0 p ··· ! 4 3! ! 4 3 ! ! 4 3 ! Figure 6.8: A pictorial rendition of the impurity ladder, where scattering occurs between two Green function lines. Impurities are inserted sequentially, and the resummed equation can be found in Eq. 6.31 of the main text.

as 1 1 X Lσ1σ2 (q, ) = L(0)σ1σ2 + σ1σa (ω ) σ4σb (ω)Lσaσ2 (q, ) , (6.31) σ4σ3 σ4σ3 p00 q p00 σbσ3 2π~N0τ V G − − G p00 where in the presentation here we have assumed an outcome — namely that in the present calculation, only momentum q and frequency ﬂow in the ladder. Note that the sum vanishes when the Green functions are not opposite causality — in that case, the poles all fall on the same side of the real axis. We will therefore specify the calculation for the case where the advanced Green function carries the momentum q; in the converse case, where the retarded Green function instead carries q, the ladder is given by the hermitian conjugate of L. In general, the ladders depend on three momenta and two frequencies, however the simpliﬁcation in this case is the result of the fact that we have presumed the presence of short-range impurities for our present purposes. We can see the fact that the other momenta fall out of the calculation by examining the kernel of the integral equation X σ1σ2 1 Aσ1σ2 Rσ3σ4 σ σ (q, ) = p q (ω ) p (ω) . (6.32) 4 3 − K V p G − G

2 2 2 The q momentum in the advanced Green function results in ξp q = ξp ~ p q/m + ~ q /2m. − − · We conﬁne ourselves to the regime p l 1 where l v τ is the mean free path due to elastic F ≡ F scattering, and note that the integral is dominated by the contribution coming from the Green function poles. Thus we set p q p q cos θ and drop the ~2 q2/2m term. Finally, we remain in the · ≈ F diﬀusive regime where τ y 1, v qτ Q 1, and v λτ Z 1, so we expand for small | | ≡ F ≡ F ≡ CHAPTER 6. PARAMAGNETO CONDUCTANCE 126

frequency and momentum to perform the angular integration. The net result for is K    2 2  1 g ιbQ ιbQ c1Z Q   − −     ιbQ g Z2h ιbQ  σ σ  1 2  1 2 (q, ) =  −  . (6.33) σ4σ3   K  ιbQ 2Z2h 1 g ιbQ   − − −    c Z2Q2 ιbQ ιbQ 1 g  1 − −

In this expression we have so-far kept the dimensionless spin-orbit coupling, Z, to all orders and introduced the functions

a g = 1 a + 1 Q2 ιa y, h = c c Q2 + ιc y . − 0 2 − 2 0 − 1 2

We have also written

1 + 2Z2 1 + 24Z4 + 32Z6 1 + 2Z2 + 8Z4 Z = , = , = , = a0 2 a1 3 a2 2 b 2 1 + 4Z 1 + 4Z2 1 + 4Z2 1 + 4Z2

1 3 + 6Z2 + 8Z4 3 + 4Z2 = , = , = . c0 2 c1 3 c2 2 1 + 4Z 1 + 4Z2 1 + 4Z2

In the converse situation, where the retarded Green function carries q, the kernel is given by the

hermitian conjugate of Eq. (6.33). In Eq. (6.33), σ1 and σ2 break the matrix into quadrants — σ1 vertically and σ horizontally — and σ and σ break each quadrant into its own 2 2 matrix. 2 3 4 × Then the ladder is given by

     G1 ιbQ ιbQ G2   −     ιbQ H H ιbQ  σ σ (0) 1 F  1 2  L 1 2 (q, ) = L [1 2π~N τ ]− =   , (6.34) σ4σ3 0 π~ τ   − K 2 N0  ιbQ H H ιbQ − 2 1 −     G ιbQ ιbQ G  2 − 1 CHAPTER 6. PARAMAGNETO CONDUCTANCE 127

14 32 G G 1 4 3 2 0 ↵ ↵ p ↵ p q (q, ✏) = ⌧ + 0 0 ! ! ✏ Figure 6.9: The spin vertex, τα, where α = x, y, z is the spin polarization, dressed by the presence of a ladder. The corrected vertex is Γ. where we have written 1 F = g + 2Z2h g c Z2Q2 4b2Q2 − 1 − 2 2 2 2 2 2 2 2 g g + 2Z h 2b Q c1Z Q g + 2Z h + 2b Q = − , = G1 2 2 G2 2 2 g + c1Z Q g + c1z Q

2 2 2 2 2 2 2 2 2 g g c1Z Q 2b Q 2Z h g c1Z Q 2b Q H = − − , H = − − . 1 g 2Z2h 2 g 2Z2h − − The diﬀusion ladder for the case where the retarded Green function carries momentum q in place

of the advanced Green function is found by = L†. L The last piece of the puzzle required to disorder average is to derive the corrected spin vertices. We correct the spin vertices by the insertion of an impurity ladder across the spin interaction as shown for one of the possible cases in Fig. 6.9. There are four possible conﬁgurations, centering around which Green function carries the momentum q and whether or not the vertex in question is an absorption (Γ) or emission (Γ˜) vertex. We use Fig. 6.9 in the situation where the advanced Green function carries q, again noting that Green functions in the sums must be opposite causality to remain non-zero, to write

X α α 1 Rσ1σ2 α Aσ3σ4 σ4σ0 Γσσ (q, ) = τσσ + p (ω)τσ σ p q (ω )Lσ σ (q, ) (6.35) 0 0 2 3 − 1 V p G G −

β β 1 X ˜ Aσ1σ2 β Rσ3σ4 σσ1 Γσσ (q, ) = τσσ + p q (ω )τσ σ p (ω)Lσ σ (q, ) . (6.36) 0 0 − 2 3 0 4 V p G − G

Both of these expressions have three potential outcomes, one each for α, β = x, y, z, the spin CHAPTER 6. PARAMAGNETO CONDUCTANCE 128

orientations. We compute the vertices to quadratic order in Z, in the diﬀusive limit, and to ﬁrst non-zero order in Q and y. The end results are

x h 2 2 2i x 2 z Γσσ (q, ) F0 1 + 3Q + 2ιy F Z τσσ 2ιQZF τ (6.37) 0 ≈ 0 0 − 0 σσ0 y h 2 2 i y Γ (q, ) F0 1 + 6Z 2Z F0 τ (6.38) σσ0 ≈ − σσ0 z h 2 2 2i z 2 x Γ (q, ) F0 1 + Q + 2ιy F Z τ + 2ιQZF τσσ (6.39) σσ0 ≈ 0 σσ0 0 0 x h 2 2 2i x 2 z Γ˜ σσ (q, ) F0 1 + 3Q + 2ιy F Z τσσ + 2ιQZF τ (6.40) 0 ≈ 0 0 0 σσ0 y h 2 2 i y Γ˜ (q, ) F0 1 + 6Z 2Z F0 τ (6.41) σσ0 ≈ − σσ0 z h 2 2 2i z 2 x Γ˜ (q, ) F0 1 + Q + 2ιy F Z τ 2ιQZF τσσ , (6.42) σσ0 ≈ 0 σσ0 − 0 0 for the case that the advanced Green function carries momentum q. We have also introduced the diﬀuson 1 F0 = . (6.43) 1 Q2 ιy 2 − In the case where the retarded Green function instead carries q, we ﬁnd that the corrected vertices are related to Eqs. (6.37)-(6.42) via hermitian conjugation. That is, the absorption (Λ) and emission

(Λ˜ ) vertices are related to Γ and Γ˜ by the relations Λ = Γ˜ † and Λ˜ = Γ† respectively.

6.6 Diamagnetic cancelation and a well behaved dc limit

In this section we demonstrate cancelation of the zero frequency terms. We ﬁrst derive expressions for the diamagnetic components and then show how they combine with the paramagnetic terms to produce a well-behaved dc limit. CHAPTER 6. PARAMAGNETO CONDUCTANCE 129

6.6.1 Diamagnetic diagrams

At ﬁrst blush, one might note that the coupling Hamiltonian Eq. (6.5) and the density commute. This indicates that in fact there is no correction to the total density due to a perturbation by the coupling Hamiltonian. However, while the total density remains unchanged, the states themselves shift in energy. Thus it is necessary to cancel oﬀ the diamagnetic contribution order by order in . As we are calculating the paramagnetic contribution to quadratic order in , we must calculate J J the diamagnetic terms to that same order. From Eq. (6.8) it is clear that in order to calculate the diamagnetic terms what is required is to calculate the density, ρ. We therefore write

X X ι R 1 + dt0 Hc(t0) − ~ cK ρ(t) = TKa†kσ(t−)akσ(t )e (6.44) h i V k σ

where a (a†) is the Fourier transformed fermion annihilation (creation) operator and the superscripts on t indicate forward (+) and backward ( ) branches of the Keldysh contour. Expanding − Eq. 6.44 to quadratic order in the coupling and Wick decomposing the result allows us to express the density as

ι X 2ν2 X Z Z ρ(t) = <,σσ(0) + J 0 dt dt τα τβ − Gk ~2 2 1 2 σ1σ2 σ3σ4 h i V kσ 4 V kq cK cK +η1,σσ1 η1η2,σ2σ3 η2 ,σ4σ αβ,η1η2 k (t t1) k q (t1 t2) k − (t2 t)χq (t1 t2) × G − G − − G − − +η ,σσ η η η ,σ σ αβ,η η 1 1 (t t ) 1−(t t) 2 2 4 3 (0)χ 1 2 (t t ) , (6.45) − Gk − 1 Gk 1 − Gq 0 1 − 2 where we have utilized the disorder averaged Green functions in order to write this expression in momentum space, and assumed all spin indices are traced over. The ﬁrst term is the usual closed fermion loop; the second and third terms are the exchange and Hartree type diagrams respectively, shown diagrammatically in Fig. 6.10. CHAPTER 6. PARAMAGNETO CONDUCTANCE 130

a) t2 b) !, k !, k q

t 0 k t , ✏, q t

✏ 2 t + 1 + t t ! !, k !, k t1 Figure 6.10: Diagrammatic representation of the 2 corrections to the diamagnetic current. Dia- J gram a) is the exchange component and diagram b) the Hartree component.

Diamagnetic exchange terms

Extracting the second term out of Eq. (6.45), expanding the Keldysh contours, and transforming to frequency space results in

2 2 Z Z ν X dω d ρ = J 0 τα τβ ex ~2 2 π π σ1σ2 σ3σ4 h i −4 V kq 2 2 h i R,σσ1 R,σ2σ3 R,σ4σ αβR αβA b() f (ω) k (ω) k q (ω ) k (ω) χq () χq () × G G − − G − h i A,σσ1 A,σ2σ3 A,σ4σ αβR αβA b() f (ω) k (ω) k q (ω ) k (ω) χq () χq () − G G − − G −

R,σσ1 R,σ2σ3 R,σ4σ αβR + f (ω) f ( ω) k (ω) k q (ω ) k (ω)χq () − G G − − G

A,σσ1 A,σ2σ3 A,σ4σ αβA + f (ω) f ( ω) k (ω) k q (ω ) k (ω)χq () − G G − − G f (ω) f ( ω) R,σσ1 (ω) A,σ2σ3 (ω ) R,σ4σ(ω)χαβR() − − Gk Gk q − Gk q − A,σσ1 R,σ2σ3 A,σ4σ αβA f (ω) f ( ω) k (ω) k q (ω ) k (ω)χq () . (6.46) − − G G − − G

These are the diamagnetic exchange terms, each of which must cancel against a paramagnetic term in the calculation in order to maintain a finite dc limit. We can extract the necessary offsetting diagrams from Eq. (6.16) — in particular, the terms that diverge in the dc limit conspire to exactly cancel ρ . We can see this cancelation by examining h exi one type of term. The fifth, seventh, and ninth terms in Eq. (6.16) written in momentum space by CHAPTER 6. PARAMAGNETO CONDUCTANCE 131

using the disorder averaged Green functions are schematically given by

Rτα Rτβ Rγ0 Rγ0 . G G G G

We can use the relationship

d 1 d 1 d k [ k]− = 0 = k [ k]− k = k (6.47) dk G G ⇒ G dk G G −dkG

to extract the diamagnetic term out of the facially divergent portions of Eqs. (6.16), (6.17), and (6.20) in the dc limit. The result is

~ Rτα Rτβ Rγ0 Rγ0 + Rτα Rγ Rτβ Rγ0 + Rγ0 Rτα Rτβ Rγ0 = Rτα Rτβ R , (6.48) G G G G G G G G G G G G −mG G G where we have used integration by parts and the fact that these terms are invariant under cyclic

~ permutations. The factor of m arises from performing the derivative of the current vertex. This shows that the self-energy and vertex correction terms conspire to exactly cancel diamagnetic exchange terms in the dc limit. That is, lines ﬁve, seven, and nine in Eqs. (6.16) and (6.17) combine with lines nine, eleven, and fourteen in Eq. (6.20) to cancel the exchange diagrams in the dc limit. The same approach can be applied to all the divergent terms in Eq. (6.16), which ultimately produces a ﬁnite dc limit when combining the diamagnetic exchange, self-energy, and vertex correction diagrams. CHAPTER 6. PARAMAGNETO CONDUCTANCE 132

Diamagnetic Hartree terms

Performing the same steps as with the diamagnetic exchange terms, the third term in Eq. (6.45) results in

2 2 Z Z ν X dω d ρ = J 0 τα τβ χαβR(0) ht ~2 2 π π σ1σ2 σ3σ4 0 h i 4 V kq 2 2 f (ω) R,σσ1 (ω) R,σ2σ(ω) <,σ4σ3 () f (ω) A,σσ1 (ω) A,σ2σ(ω) <,σ4σ3 () . (6.49) × Gk Gk Gq − Gk Gk Gq

These terms, too, must meet associated paramagnetic terms that exactly cancel them in order to maintain a ﬁnite dc limit. Equations (6.25) and (6.26) contain two terms each that a priori do not display a well-behaved dc limit — these are the terms we must examine to extract diamagnetic cancelation. Approaching the question schematically as in the case of the diamagnetic exchange diagram, we examine the second lines in Eqs. (6.25) and (6.26) to write

~ Rτα Rγ0 Rγ0 <τβ + Rγ0 Rτα Rγ0 <τβ = Rτα R <τβ . (6.50) G G G G G G G G −mG G G

Thus, expanding the second line of Eq. (6.25) using the identity provided in Eq. (6.47) shows that schematically similar terms in Eq. (6.25) and (6.26) conspire to remove the diamagnetic contribution in the dc limit. The same approach can be used for the third line of Eq. (6.25), all of which conspires to produce a ﬁnite dc limit for the Hartree terms as well, mirroring the result of the previous subsection for the exchange terms.

6.7 The Conductivity

In this section we will produce our result for the conductivity to second order in the spin-orbit coupling parameter. We begin with a close examination of the self-energy terms in order to demonstrate our method of disorder averaging and calculating the full array of contributing diagrams. We CHAPTER 6. PARAMAGNETO CONDUCTANCE 133

will show that the diagrams admitting maximal ladder insertions are dominant, and therefore focus on those terms in our presentation of the final results. When disorder averaging we first account for disorder along a single electron line by the replacement of bare Green functions, G, with their disorder averaged counterparts, . This restores G translational invariance and therefore allows us to transform to momentum space. We then admit the maximal number of impurity ladders allowed within a diagram without crossing impurity lines, recalling that it is only possible to insert a ladder between Green functions of opposite causality. This accounts for interference effects between electron lines. Finally, we discuss the approximations used in computing the integrals. We perform the momentum sums via the replacement

Z Z 2π Z Z 2π 1 X ∞ dθ ∞ dθ = N0 dξk N0 dξk V µ 0 2π ≈ 0 2π k − −∞

m where N0 = 2π~2 is the 2d density of states of the metal. The Fermi surface, µ, is the highest energy

scale in the system, and so we extend the lower bound of the ξk integral to negative inﬁnity. This allows us to shift the frequency ω out of our Green functions, and so we set ω 0 in all Green → functions. The maximum value of is set by the bandwidth of the quantum magnet, while the maximum value of Ω is set by instrumental limitations. The scattering time in the typical metal is

14 on the order of τ 10− s, so that τ 1 and Ωτ 1, and therefore we also set 0 and ∼ → Ω 0 in the Green functions. As a result, Green functions carry no frequency arguments in our → calculation, and the ω integrals become straightforward. A general result we will utilize in this vein is ! ! Z dω f (ω) f ( ω) = coth 1 F() , (6.51) − 2 2kBT − ≡

where T is the temperature and kB is the Boltzmann constant. We approximate the current vertices

at the Fermi surface such that γσσ0 v cos θδ , and, since Green functions become simply k ≈ F σσ0

functions of ξk due to the presence of the diﬀusive pole, all the Green functions have the same CHAPTER 6. PARAMAGNETO CONDUCTANCE 134

frequency and momentum arguments. Thus, momentum integrals over Green functions may be calculated using contour methods.

6.7.1 Self-Energy and Vertex Contributions

At this point we will depict the simpliﬁed results for the diagrams given the approximations discussed above and the cancelation arising from combining paramagnetic and diamagnetic terms. The ﬁrst outcome is to realize that, since Green functions now contain no frequency arguments and identical momentum arguments, momentum sums over multiple Green functions of the same causality, either retarded or advanced, vanish due to the presence of poles on only one side of the real axis. Thus, we drop all R R R R and A A A A type terms. We also account for the G G G G G G G G diamagnetic cancelation arising between the self-energy and vertex correction by combining bare dc terms of Eq. (6.16) and Eq. (6.17) with diamagnetic terms to produce expressions that make it clear that the dc limit is well behaved. That is to say, the twelfth term of Eq. (6.16) can now be rewritten using Eq. (6.47) such that

~ Aτα Rτβ Aγ0 Aγ0 = Rτα Aτβ R Aτα Rγ Rτβ Aγ0 Aγ0 Aτα Rτβ Aγ0 , G G G G −mG G G − G G G G − G G G G meaning this term produces diamagnetic cancelation, and additionally terms that balance the twelfth line of Eq. (6.17) and the sixteenth line of Eq. (6.20). Similarly, we can use Eq. (6.47) to rewrite the eighth line of Eq. (6.17) such that

~ Rγ0 Rτα Aτβ Rγ0 = Aτα Rτβ A Rτα Aγ Aτβ Rγ0 Rτα Aτβ Rγ0 Rγ0 , G G G G −mG G G − G G G G − G G G G which results in cancelation of the A R A diamagnetic term, in addition to producing terms that G G G balance the eighth line of Eq. (6.16) and the twelfth line of Eq. (6.20) to explicitly display a ﬁnite CHAPTER 6. PARAMAGNETO CONDUCTANCE 135

dc limit. The net results for the self-energy diagrams is

2 2 2 Z ν e X X d δΠR,(a)(Ω) = J 0 γσ000σ00 γσσ0 τα τβ xx ~3 2 π 2 k k σ1σ2 σ3σ4 − 4 V αβ qk (2 ) ( Ωb() R,σ00σ1 R,σ2σ3 R,σ4σ A,σ0σ000 χαβR() × − Gk Gk Gk Gk q

+ Ωb() R,σ00σ1 R,σ2σ3 R,σ4σ A,σ0σ000 χαβA() Gk Gk Gk Gk q + [F() F( Ω)] R,σ00σ1 R,σ2σ3 R,σ4σ A,σ0σ000 χαβR() − − Gk Gk Gk Gk q [F() F( Ω)] R,σ00σ1 A,σ2σ3 R,σ4σ A,σ0σ000 χαβR() − − − Gk Gk Gk Gk q ) + [F() F( Ω)] R,σ00σ1 A,σ2σ3 R,σ4σ R,σ0σ000 χαβR() (6.52) − − Gk Gk Gk Gk q for the conﬁguration with the self-energy bubble on the top branch, and

2 2 2 Z ν e X X d δΠR,(b)(Ω) = J 0 γσ000σ00 γσσ0 τα τβ xx ~3 2 π 2 k k σ1σ2 σ3σ4 − 4 V αβ qk (2 ) ( Ωb() R,σ00σ A,σσ1 A,σ2σ3 A,σ4σ000 χαβR() × − Gk Gk Gk Gk q

+ Ωb() R,σ00σ A,σσ1 A,σ2σ3 A,σ4σ000 χαβA() Gk Gk Gk Gk q + [F() F( Ω)] R,σ00σ A,σσ1 A,σ2σ3 A,σ4σ000 χαβA() − − Gk Gk Gk Gk q + [F( + Ω) F()] R,σ00σ A,σσ1 R,σ2σ3 A,σ4σ000 χαβA() − Gk Gk Gk Gk q ) [F( + Ω) F()] A,σ00σ A,σσ1 R,σ2σ3 A,σ4σ000 χαβA() (6.53) − − Gk Gk Gk Gk q for the conﬁguration with the self-energy bubble on the bottom. Note that we have solved the ω integrals in producing each of these expressions. The vertex correction given by Eq. (6.20) also acquires dc terms from this approach. The result, which again conspires to balance the facially CHAPTER 6. PARAMAGNETO CONDUCTANCE 136

divergent terms, is:

2 2 2 Z ν e X X d δΠR,(c)(Ω) = J 0 γσ000σ00 γσσ0 τα τβ xx ~3 2 π 2 k k σ1σ2 σ3σ4 − 4 V αβ qk (2 ) ( Ωb() R,σ00σ1 R,σ2σ A,σ0σ3 A,σ4σ000 χαβR() × − Gk Gk Gk Gk q

+ Ωb() R,σ00σ1 R,σ2σ A,σ0σ3 A,σ4σ000 χαβA() Gk Gk Gk Gk q [F() F( Ω)] R,σ00σ1 R,σ2σ A,σ0σ3 R,σ4σ000 χαβR() − − − Gk Gk Gk Gk q [F() F( Ω)] R,σ00σ1 R,σ2σ A,σ0σ3 A,σ4σ000 χαβR() − − − Gk Gk Gk Gk q [F() F( Ω)] R,σ00σ1 A,σ2σ A,σ0σ3 A,σ4σ000 χαβR() − − − Gk Gk Gk Gk q + [F( + Ω) F()] A,σ00σ1 R,σ2σ A,σ0σ3 A,σ4σ000 χαβA() − Gk Gk Gk Gk q [F( + Ω) F()] R,σ00σ1 R,σ2σ A,σ0σ3 A,σ4σ000 χαβA() − − Gk Gk Gk Gk q + [F( + Ω) F()] R,σ00σ1 R,σ2σ R,σ0σ3 A,σ4σ000 χαβA() − Gk Gk Gk Gk q + [F() F( Ω)] R,σ00σ1 A,σ2σ A,σ0σ3 R,σ4σ000 χαβR() − − Gk Gk Gk Gk q ) [F( + Ω) F()] A,σ00σ1 R,σ2σ R,σ0σ3 A,σ4σ000 χαβA() . (6.54) − − Gk Gk Gk Gk q

We move now to inserting ladders. First note that the corrected spin vertices, given by Eqs. (6.37) - (6.42), account for the bare vertex, as can be seen by Eqs. (6.35) and (6.36). By contrast, the ladder functions do not. Therefore, when inserting ladders, in order to consider all possible terms, it is necessary to evaluate those permitting cross-bubble ladders both with and without the ladder separately. The first set of terms to consider are those permitting only a single cross-bubble ladder. Those are the first, second, and third lines of Eqs. (6.52) and (6.53), and the first, second, fourth, and seventh lines of Eq. (6.54) — Fig. 6.11 provides a visual of what it means to insert a ladder into a diagram. Green functions in the expressions conform to the diagram in order, from left to right, beginning at the left-most current vertex and wrapping around the diagram against the depicted CHAPTER 6. PARAMAGNETO CONDUCTANCE 137

✏ q k q ! ! ✏ ⌦ + ⌦ k k0 ! + ✏ k k 0 ! ! q k q k0 q q ✏ 5 q k k0 ✏ ! ! k ! ⌦ k0 k0 Figure 6.11: The terms emerging from Eq. eqrefse and Eq. (6.54), for χR, that admit only the insertion of a single cross-bubble ladder. The ladder is highlighted in grey, and energy and momentum ﬂow around the bubble are depicted. Note that inserting the ladder breaks the bubble into two separate momentum sums, one for k and the other for k0, by accounting for elastic scattering against impurities.

ﬂow from top to bottom, i.e., clockwise. Extracting the terms in question, we have

R,σ σ R,σ σ β R,σ σ A,σ σ 00 1 τα 2 3 τ 4 γσσ0 0 000 γσ000σ00 RRRAa k σ1σ2 k q σ3σ4 k k k k ≡ G G − G G R,σ σ A,σ σ A,σ σ β A,σ σ 00 γσσ0 0 1 τα 2 3 τ 4 000 γσ000σ00 RAAAb k k k σ1σ2 k q σ3σ4 k k ≡ G G G − G R,σ σ R,σ σ A,σ σ β A,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 , RRAAc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G where the overline indicates disorder averaging, and we have adopted the shorthand of listing the Green functions in order around the bubble clockwise as a notational convenience. In this case, each term is associated with both χR and χA. These three terms can accommodate the insertion of a single cross-bubble ladder, but as mentioned previously we must additionally calculate the situation in which no ladders are inserted. In what follows, we will explicitly calculate the χR terms, noting that the χA case proceeds identically. CHAPTER 6. PARAMAGNETO CONDUCTANCE 138

Zero ladders

In the absence of cross-bubble ladders, we write

(0) 1 X = R,σ00σ1 τα R,σ2σ3 τβ R,σ4σγσσ0 A,σ0σ000 γσ000σ00 RRRAa k σ1σ2 k σ3σ4 k k k k V k G G G G (0) 1 X = R,σ00σ1 τα A,σ2σ3 τβ A,σ4σγσσ0 A,σ0σ000 γσ000σ00 RAAAb k σ1σ2 k σ3σ4 k k k k V k G G G G (0) 1 X = R,σ00σ1 τα R,σ2σγσσ0 A,σ0σ3 τβ A,σ4σ000 γσ000σ00 , RRAAc k σ1σ2 k k k σ3σ4 k k V k G G G G where in this case, without ladder insertions, the disorder average is accomplished by utilizing disorder averaged Green functions and integrating out the momentum. Performing the sums allows us to arrive at the expressions

(0) RRRA = 2πN v2 τ3~ χxxR() + χyyR() + χzzR() Z2 3χxxR() + χyyR() + 4χzzR() a − 0 F q q q − q q q (0) RAAA = 2πN v2 τ3~ χxxR() + χyyR() + χzzR() Z2 3χxxR() + χyyR() + 4χzzR() b − 0 F q q q − q q q (0) RRAA = 4πN v2 τ3~ χxxR() + χyyR() + χzzR() Z2 3χxxR() + χyyR() + 4χzzR() , c 0 F q q q − q q q

R and recall that the unitless spin-orbit coupling parameter Z = vFλτ. The χ portion of the zero- ladder contribution is then proportional to the sum of these two results. We thus ﬁnd, after solving the case proportional to the advanced susceptibility as well, that the total zero-ladder term arising due to the self-energy and vertex corrections becomes

2 2 2 2 3 Z ν e v τ N0 X d δΠR(0)(Ω) = J 0 F xx ~2 π − 4 V q 2 ( [F() F( Ω)] χxxR() + χyyR() + χzzR() Z2 3χxxR() + χyyR() + 4χzzR() × − − q q q − q q q ) [F( + Ω) F()] χxxA() + χyyA() + χzzA() Z2 3χxxA() + χyyA() + 4χzzA() . (6.55) − − q q q − q q q CHAPTER 6. PARAMAGNETO CONDUCTANCE 139

One cross-bubble ladder

The one-ladder terms are given by the same diagrams as the zero-ladder case, only now we include the presence of a single cross-bubble ladder, as depicted in Fig. 6.11. A complication arises

because there are two independent ways to insert cross-bubble ladders in the RRAAc diagram — either across the bubble from top right to bottom left, as shown in Fig. 6.11, or top left to bottom right. Thus there are four terms to calculate in the case of diagrams proportional to the retarded susceptibility:

(1) X 1 A,σcσ000 σ000σ00 R,σ00σ1 α R,σ2σa σaσb R,σbσ3 β R,σ4σ σσ0 A,σ0σd RRRA = γ τσ σ σ σ (q, + Ω) τσ σ γ a V2 Gk k Gk 1 2 Gk L c d Gk0 3 4 Gk0 k0 Gk0 kk0 (1) X 1 R,σcσ σσ0 A,σ0σ1 α A,σ2σa σaσb A,σbσ3 β A,σ4σ00 σ00σ000 R,σ000σd RAAA = γ τσ σ Lσ σ (q, Ω) τσ σ γ b V2 Gk k Gk 1 2 Gk c d − Gk0 3 4 Gk0 k0 Gk0 kk0 (1a) X 1 R,σcσ1 α R,σ2σ σσ0 A,σ0σa σaσb A,σbσ3 β A,σ4σ000 σ000σ00 R,σ00σd RRAA = τσ σ γ Lσ σ (q, + Ω) τσ σ γ c V2 Gk 1 2 Gk k Gk c d Gk0 3 4 Gk0 k0 Gk0 kk0 (1b) X 1 A,σcσ000 σ000σ00 R,σ00σ1 α R,σ2σa σaσb R,σbσ σσ0 A,σ0σ3 β A,σ4σd RRAA = γ τσ σ σ σ (q, Ω) γ τσ σ . c V2 Gk k Gk 1 2 Gk L c d − Gk0 k0 Gk0 3 4 Gk0 kk0

We see here how the insertion of ladders proceeds. A cross-bubble ladder insertion disassociates the two halves of the bubble, causing the two current vertices to decohere; then sums over each momentum arise — namely the momenta ﬂowing around each subdivided section. Solving these equations to Z2 order, we ﬁnd

χxxR + χzzR (1) 2 2 3 q ( ) q ( ) RRRA = 4πN0v Z τ ~ a F 1 Q2 + ι(y Ωτ) 2 − χxxR + χzzR (1) 2 2 3 q ( ) q ( ) RAAA = 4πN0v Z τ ~ b F 1 Q2 ι(y + Ωτ) 2 − χxxR + χzzR (1a) 2 2 3 q ( ) q ( ) RRAA = 4πN0v Z τ ~ c − F 1 Q2 ι(y + Ωτ) 2 − χxxR + χzzR (1b) 2 2 3 q ( ) q ( ) RRAA = 4πN0v Z τ ~ , c − F 1 Q2 + ι(y Ωτ) 2 − CHAPTER 6. PARAMAGNETO CONDUCTANCE 140

k k q k k q ⌦ ! + ⌦ ! + ! + ⌦ ! + ⌦ ✏ ✏ q ✏ q ✏ ✏ ! ✏ ! ! ! q q k k k k

Figure 6.12: Diagrams in which it is possible to insert a corrected spin vertex in addition to a cross- bubble ladder. The ladder is depicted in grey, and triangles over spin vertices indicate a corrected vertex.

where (1a) has the ladder crossing left to right, and (1b) the reverse. Also recall that Q = vFqτ and y = τ . The diﬀuson pole enters through the ladder and diverges as q, 0, indicating that most | | → of the contribution arises for q and near zero.

One corrected spin vertex

A second way to insert a single ladder is when it is possible to correct a single spin vertex in the absence of a cross-bubble ladder. Figure 6.12 depicts the arrangements where this may be accomplished, and we recall that one must account for the diagrams both with and without a cross- bubble ladder inserted. From Eq. (6.54) we extract the third, ﬁfth, sixth, and eighth lines. That is, we must calculate

R,σ σ R,σ σ A,σ σ β R,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 RRARc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G R,σ σ A,σ σ A,σ σ β A,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 RAAAc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G A,σ σ R,σ σ A,σ σ β A,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 ARAAc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G R,σ σ R,σ σ R,σ σ β A,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 RRRAc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G averaged with one corrected spin vertex and no cross-bubble ladder. The ﬁrst two terms above are those attached to the retarded susceptibility, while the last two are attached to the advanced susceptibility. In keeping with the general approach taken in this section, we calculate the ﬁrst two CHAPTER 6. PARAMAGNETO CONDUCTANCE 141

explicitly. As with cross-bubble ladders, emplacing corrected vertices is only possible when connecting Green functions of opposite causality. Thus, we write

(10) 1 X = R,σ00σ1 τα R,σ2σγσσ0 A,σ0σ3 Γ˜ β q, R,σ4σ000 γσ000σ00 RRARc k σ1σ2 k k k σ3σ4 ( ) k k V k G G G G

(10) 1 X = R,σ00σ1 Γα q, A,σ2σγσσ0 A,σ0σ3 τβ A,σ4σ000 γσ000σ00 , RAAAc k σ1σ2 ( ) k k k σ3σ4 k k V k G G G G and we note a key difference: the first line admits an emission vertex, and the second an absorption vertex. Additionally, it is important to understand that inserting corrected current vertices also requires connecting Green functions of opposite causality. However, the off-diagonal elements of the uncorrected current vertices result in terms suppressed in the diffuse limit, i.e., suppressed by k l 1, and so we neglect the anomalous portions of the current vertices and utilize corrected F current vertices throughout. We solve these two diagrams to second order in the unitless spin-orbit coupling Z and find

2 3 2 3 (10) 2πN0vFτ ~ h xxR yyR zzR i 4πιN0vF QZτ h xzR zxR i RRARc = χq () + χq () + χq () + χq () χq () − 1 2 1 2 − 2 Q ιy 2 ι − 2 Q y " − # 2πN v2 Z2τ3~ 0 F 2 + ι χxxR 2 ι χyyR + 2 + ι χzzR 3 3Q 2 y q ( ) Q 2 y q ( ) 2Q 4 y q ( ) − 1 Q2 ιy − − 2 −

2 3 2 3 (10) 2πN0vFτ ~ h xxR yyR zzR i 4πιN0vF QZτ h xzR zxR i RAAAc = χq () + χq () + χq () + χq () χq () − 1 2 1 2 − 2 Q ιy 2 ι − 2 Q y " − # 2πN v2 Z2τ3~ 0 F 2 + ι χxxR 2 ι χyyR + 2 + ι χzzR , 3 3Q 2 y q ( ) Q 2 y q ( ) 2Q 4 y q ( ) − 1 Q2 ιy − − 2 −

and so we see both terms evaluate to the same quantity. The total one-ladder expression to second order in the unitless spin-orbit coupling term Z is CHAPTER 6. PARAMAGNETO CONDUCTANCE 142 the result of combining both the single cross-bubble ladder terms and the single vertex correction terms. The result is

2 2 2 2 3 Z ( " xxR yyR zzR ν e v τ N0 X d χq () + χq () + χq () δΠR(1)(Ω) = J 0 F [F() F( Ω)] xx − 2~2V 2π − − 1 2 q 2 Q ιy − xzR zxR 2 2 xxR 2 yyR 2 zzR # 2ιQZ χq () χq () Z 3Q + 2ιy χq () Q 2ιy χq () + 2Q + 4ιy χq () − + − − 2 3 − 1 2 1 2 2 Q ιy 2 Q ιy − − " xxA yyA zzA xzA zxA χq () + χq () + χq () 2ιQZ χq () χq () + Ω − [F( ) F( )] 1 2 − − Q2 + ιy − 1 2 2 2 Q + ιy 2 2 xxA 2 yyA 2 zzA #) Z 3Q 2ιy χq () Q + 2ιy χq () + 2Q 4ιy χq () + − − − , 3 (6.56) 1 2 2 Q + ιy where we have included the portion dependent on the advanced spin susceptibility as well.

One corrected spin vertex and a cross-bubble ladder

As depicted, the diagrams presented in Fig. 6.12 can also accommodate the insertion of a cross- bubble ladder in addition to a single corrected spin vertex. Emplacing both a cross-bubble ladder and a corrected spin vertex results in the expressions

(20) X 1 R,σcσ1 α R,σ2σ σσ0 A,σ0σa σaσb A,σbσ3 β R,σ4σ000 σ000σ00 R,σ00σd RRAR = τσ σ γ Lσ σ (q, + Ω) Γ˜ σ σ (q, ) γ vc V2 Gk 1 2 Gk k Gk c d Gk0 3 4 Gk0 k0 Gk0 kk0

(20) X 1 R,σcσ1 α R,σ2σ σσ0 A,σ0σa σaσb A,σbσ3 β R,σ4σ000 σ000σ00 R,σ00σd RAAA = Γσ σ γ Lσ σ (q, + Ω) τσ σ (q, ) γ . vc V2 Gk 1 2 Gk k Gk c d Gk0 3 4 Gk0 k0 Gk0 kk0

For these diagrams, there is only one way to emplace a ladder — bottom right to top left, connecting an advanced and a retarded Green function. As a result, the same energy is ﬂowing in the ladder in both instances, and both diagrams evaluate to the same quantity:

π 2 2τ3~ χxxR + χzzR (20) (20) 8 N0vFZ q ( ) q ( ) RRARc = RAAAc = . 1 Q2 ιy 1 Q2 (ιy + Ωτ) 2 − 2 − CHAPTER 6. PARAMAGNETO CONDUCTANCE 143

✏ k q q ! ✏ k k k0 ! k ⌦ + ⌦ 0 ! ! + q ! ✏ k q k0 q q ✏ ! 5 ✏ k ! q k k0 0 ! ⌦ k0 k Figure 6.13: Diagrams that can sustain the insertion of two corrected spin vertices in addition to a cross-bubble ladder. The cross-bubble ladder is depicted in grey, and the corrected spin vertices appear as triangles around spin interactions. Cross-bubble ladders break the bubble up into two halves, each with its own momentum, eﬀectively decohering the charge vertices.

Two corrected spin vertices

Diagrams that can accommodate the insertion of two corrected spin vertices and a cross-bubble ladder are depicted in Fig. 6.13. We note for a final time that each of these types of diagrams must be examined with and without the cross-bubble ladder insertion by virtue of Eq. (6.31) and how the cross-bubble ladder is defined. The terms we must consider at this point are the fourth, and fifth lines of Eqs. (6.52) and (6.53), and the ninth and tenth terms in Eq. (6.54):

R,σ σ A,σ σ β R,σ σ A,σ σ 00 1 τα 2 3 τ 4 γσσ0 0 000 γσ000σ00 RARAa k σ1σ2 k q σ3σ4 k k k k ≡ G G − G G R,σ σ A,σ σ β R,σ σ R,σ σ 00 1 τα 2 3 τ 4 γσσ0 0 000 γσ000σ00 RARRa k σ1σ2 k q σ3σ4 k k k k ≡ G G − G G R,σ σ A,σ σ A,σ σ β R,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 RAARc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G R,σ σ A,σ σ R,σ σ β A,σ σ 00 γσσ0 0 1 τα 2 3 τ 4 000 γσ000σ00 RARAb k k k σ1σ2 k q σ3σ4 k k ≡ G G G − G A,σ σ A,σ σ R,σ σ β A,σ σ 00 γσσ0 0 1 τα 2 3 τ 4 000 γσ000σ00 ARAAb k k k σ1σ2 k q σ3σ4 k k ≡ G G G − G A,σ σ R,σ σ R,σ σ β A,σ σ 00 1 τα 2 γσσ0 0 3 τ 4 000 γσ000σ00 , ARRAc k σ1σ2 k q k k q σ3σ4 k k ≡ G G − G − G where the ﬁrst three terms are associated with the retarded susceptibility, and the remaining three are associated with the advanced susceptibility. Note here that two self-energy RARA terms arise, one of which corresponds to the (a) diagram and the other of which corresponds to the (b) diagram. CHAPTER 6. PARAMAGNETO CONDUCTANCE 144

Following the same outline as what has come before, we will calculate the terms associated with the retarded susceptibility explicitly, while reminding that the other set of terms proceed identically. We ﬁrst perform the case with no cross-bubble ladder. This requires the insertion of two corrected spin vertices. Additionally, a wrinkle arises in the case of the RARA terms, namely the existence of two related diagrams that contribute at the Z2 order. From Eq. (6.31) we can see that, while the sum over two retarded or advanced Green functions vanishes, a single impurity can connect two Green functions of the same causality. Thus, we must consider two additional diagrams derived from the term in question and illustrated in Fig. 6.14. These diagrams arise in this case and not any others due to the positioning of the Green functions around the bubble — as we have seen, an impurity scattering event breaks the bubble into two momentum sums, both of which require poles on either side of the real axis to remain non-zero. Only the RARA self-energy type diagrams can fulﬁll this requirement. The expressions to disorder average are therefore given by

1 X = R,σ00σ1 Γα q, A,σ2σ3 Γ˜ β q, R,σ4σγσσ0 A,σ0σ000 γσ000σ00 RARAa k σ1σ2 ( ) k σ3σ4 ( ) k k k k V k G G G G X 1 A,σcσ000 σ000σ00 R,σ00σ1 α A,σ2σa (0)σaσb A,σbσ3 β R,σ4σ σσ0 A,σ0σd + γ Γσ σ (q, ) Lσ σ Γ˜ σ σ (q, ) γ V2 Gk k Gk 1 2 Gk c d Gk0 3 4 Gk0 k0 Gk0 kk0 X 1 R,σcσ σσ0 A,σ0σ00 σ00σ000 R,σ000σa (0)σaσb R,σbσ1 α A,σ2σ3 β R,σ4σd + γ γ Lσ σ Γσ σ (q, ) Γ˜ σ σ (q, ) V2 Gk k Gk k Gk c d Gk0 1 2 Gk0 3 4 Gk0 kk0 (2) 1 X = R,σ00σγσσ0 R,σ0σ1 Γα q, A,σ2σ3 Γ˜ β q, R,σ4σ000 γσ000σ00 RARRa k k k σ1σ2 ( ) k σ3σ4 ( ) k k V k G G G G (2) 1 X = R,σ00σ1 Γα q, A,σ2σγσσ0 A,σ0σ3 Γ˜ β q, R,σ4σ000 γσ000σ00 . RAARc k σ1σ2 ( ) k k k σ3σ4 ( ) k k V k G G G G

(0)σaσb The single scattering event is characterized by the bare Lσcσd which is diagonal in both sets of spin indices. After the summations, and including the signs of these terms in Eqs. (6.52) and (6.54), CHAPTER 6. PARAMAGNETO CONDUCTANCE 145

these diagrams combine to give the result

2 2 3 (2) (2) 4πN v Z τ ~ h i = 0 F χxxR + χyyR + χzzR . RARAa RARRa RAARc 2 7 q ( ) 4 q ( ) 11 q ( ) − − 1 Q2 ιy 2 −

Note the emergence of two orders of the diﬀusive pole here, arising as a result of insertion of the corrected spin vertices. Producing the entire two-ladder term — either a cross-bubble ladder with a single corrected spin vertex, or two corrected spin vertices — then requires the addition of this

(20) (20) result with the RRARc and RAAAc diagrams. The net outcome, including the portion dependent on the advanced susceptibility, becomes

2 2 2 2 2 3 Z " xxR zzR ν e v Z τ N X 4 [F() F( Ω)] χq () + χq () R,(2) 0 F 0 d − − δΠxx (Ω) = J 2~2V 2π 1 Q2 ιy 1 Q2 ι (y + Ωτ) q 2 − 2 − xxA zzA xxR yyR zzR 4 [F( + Ω) F()] χq () + χq () [F() F( Ω)] 7χq () + 4χq () + 11χq () − + − − 2 − 1 Q2 + ιy 1 Q2 + ι (y Ωτ) 1 Q2 ιy 2 2 − 2 − xxA yyA zzA # [F( + Ω) F()] 7χq () + 4χq () + 11χq () − . 2 (6.57) − 1 2 2 Q + ιy

Three ladder diagrams

The diagrams that can accommodate two corrected spin vertices and a cross-bubble ladder are

RARRa, RAARc, ARAAb, and ARRAc. The ﬁrst two are associated with the retarded susceptibility, and the latter two are associated with the advanced susceptibility. We will demonstrate the calculation for the retarded susceptibility and then present both in the end. The diagrams to calculate, CHAPTER 6. PARAMAGNETO CONDUCTANCE 146

✏ ✏ q q ! ✏ ! ✏ k k k k0 k q ! ! ! k q k0 q ! 1 RARAse = + ! ⌦ ✏ k k0 k ! ⌦ q k0 ! ✏ k0 k k0 q k ! ! + ! ⌦ k

Figure 6.14: Two additional diagrams arise out of the RARAa, and similarly RARAb, term due to the ability to connect the two retarded and two advanced Green functions with a single impurity scattering event, denoted by the star connected to a dotted line. These situations are possible in this diagram and not the others due to the fact that this particular arrangement of Green functions maintains two separate non-zero sums after a scattering event.

expressed explicitly as sums, are

(3) 1 X RARR = Rσcσ00 γk Rσ000σ1 Γα q, Aσ2σa Lσaσb q, Ω a k σ00σ000 k σ1σ2 ( ) k σcσd ( ) V k G G G − X 1 Aσbσ3 β Rσ4σ k0 Rσ0σd Γ˜ σ σ (q, ) γσσ × V Gk0 3 4 Gk0 0 Gk0 k0 (3a) 1 X RAAR = Rσcσ00 γk Rσ000σ1 Γα q, Aσ2σa Lσaσb q, Ω c k σ00σ000 k σ1σ2 ( ) k σcσd ( ) V k G G G − X 1 Aσbσ k0 Aσ0σ3 β Rσ4σd γσσ Γ˜ σ σ (q, ) × V Gk0 0 Gk0 3 4 Gk0 k0 (3b) 1 X RAAR = Rσcσ1 Γα q, Aσ2σγk Aσ0σa Lσaσb q, + Ω c k σ1σ2 ( ) k σσ0 k σcσd ( ) V k G G G X 1 Aσbσ3 β Rσ4σ00 k0 Rσ000σd Γ˜ σ σ (q, ) γσ σ . × V Gk0 3 4 Gk0 00 000 Gk0 k0

Note here the two potential cross-bubble ladder insertions associated with the RAARc terms: either right to left across the bubble (3a) or left to right (3b). Additionally, each term can take two corrected spin vertices as depicted in Fig. 6.13. We calculate this expression to second order in Z CHAPTER 6. PARAMAGNETO CONDUCTANCE 147

as well, with the outcome

2 2 3 xxR zzR (3) (3a) (3b) 16πN0v Z τ ~ χq () + χq () + + = F . RARRa RAARc RAARc 2 − 1 Q2 ιy 1 Q2 ι (y + Ωτ) 2 − 2 −

Then, the full three-ladder term, including both the sections dependent on the retarded and advanced QSL susceptibilities is

2 2 2 2 2 3 Z ( xxR zzR ν e v Z τ N X [F() F( Ω)] χq () + χq () R,(3) 0 F 0 d − − δΠxx (Ω) = 2J ~2 π 1 2 1 V q 2 2 2 2 Q ιy 2 Q ι (y + Ωτ) − − xxA zzA ) [F( + Ω) F()] χq () + χq () − . 2 (6.58) − 1 Q2 + ιy 1 Q2 + ι (y Ωτ) 2 2 −

R,(3) Comparing Eqs. (6.55), (6.56), (6.57), and (6.58), we can see that δΠxx (Ω) is the dominant term. The unitless spin-orbit coupling Z 1 and the unitless momentum Q compete in each term. Ladders diverge as 1/Q2, so we examine powers of Z2 and 1/Q2. The zero-ladder term goes as δΠR,(0)(Ω) 1, while the most divergent one-ladder terms go as δΠR,(1)(Ω) Z2/Q4. Likewise, xx ∼ xx ∼ the two-ladder terms also diverge as δΠR,(2)(Ω) Z2/Q4. The three-ladder terms, by contrast, xx ∼ diverge as δΠR,(3)(Ω) Z2/Q6. Thus, we focus on terms with the most ladders possible in our xx ∼ examination of the case without spin-orbit coupling, i.e., Z 0, and eventually our presentation → of the conductivity, as maximal ladder terms are dominant.

6.7.2 The Z 0 Limit → In this section we examine the Z 0 limit, where spin-orbit coupling drops out of the calculation. → In this limit, the ladders become diagonal:

1 σ1σ2 1 2 − L (q, ) = (2πN τ~)− Dq τ ιτ δσ σ δσ σ , (6.59) σ3σ4 0 − 1 2 3 4 CHAPTER 6. PARAMAGNETO CONDUCTANCE 148

and the converse case remains = L†. Without spin-orbit coupling the spin vertices also simplify, L resulting in 1 Γα = Dq2τ ιτ − τα (6.60) σσ0 − σσ0

for both absorption and emission vertices, and the partner vertices with the retarded and advanced

Green functions reversed remain Λ = Γ†. We know from the above that diagrams that can accommodate three ladders are the most contributive due to the presence of three powers of the diﬀusive (3) (3a) (3b) pole. Thus, in the absence of spin-orbit coupling, we consider the RARRa , RAARc , and RAARc diagrams once more. By expanding each momentum sum for Q 1, we ﬁnd

2 2 3 xxR yyR zzR (3) (3a) (3b) 4πN0v Q τ ~ χq () + χq () + χq () + + = F . RARRa RAARc RAARc 2 − 1 Q2 ιy 1 Q2 ι (y + Ωτ) 2 − 2 −

Note that Q eﬀectively takes the place of Z in the numerator, and χyy now arises in the calculation. With spin-orbit coupling the χyy component of the susceptibility drops out of the calculation because the measurement axis is the x axis, and the spin ﬂuctuations are polarized along the z axis. Setting Z 0 recovers the symmetry in the spin sector. → The Hartree and Barbell diagrams vanish identically in the absence of spin-orbit coupling. This is easily seen because the spin vertices are now traceless, which results in vanishing sums over all spin indices in a fermion bubble that only contains a single spin vertex. The Hartree and Barbell diagrams each have two fermion bubbles with only a single spin vertex in each of them, both of which therefore vanish. Thus, these diagrams play no role in the conductivity calculation in the absence of spin-orbit coupling. The total retarded response function in the absence of spin-orbit CHAPTER 6. PARAMAGNETO CONDUCTANCE 149

coupling is therefore given by

2 2 2 2 3 Z ( xxR yyR zzR ν e v τ N X [F() F( Ω)] χq () + χq () + χq () R,(3q) 0 F 0 2 d − − δΠxx (Ω) = J Q ~2 π 1 2 1 2 V q 2 2 2 2 Q ιy 2 Q ι (y + Ωτ) − − xxA yyA zzA ) [F( + Ω) F()] χq () + χq () + χq () − . 2 (6.61) − 1 Q2 + ιy 1 Q2 + ι (y Ωτ) 2 2 − 6.7.3 The conductivity

The zeroth order calculation

We call the conductivity to zeroth order in the zeroth order calculation. That is, from Eq. (6.6) J we have ι D E ηη0 p η p η0 Πxx (p, t) = TK jp(t ) j p(0 ) . (6.62) −~ −

We set the external momentum to zero, as required by Eq. (6.12), and calculate

2 Z ι e X dω σσ 0,σ σ σ σ σ σ ΠR (Ω) = γ 0 γ 00 000 f (ω) f (ω Ω) R 0 00 (ω) A 000 (ω Ω) xx ~ π k k k k V k 2 − − G G − + f (ω Ω) Rσ0σ00 (ω) Rσ000σ(ω Ω) f (ω) Aσ0σ00 (ω) Aσ000σ(ω Ω) , (6.63) − Gk Gk − − G Gk − where note that we have one corrected (γ) and one uncorrected (γ0) charge vertex so as not to double-count impurity scattering events. Using Eq. (6.47) on the last two terms in the dc limit, we write

2 Z ι e X dω σ σ 0,σ σ σ σ σσ ΠR (Ω) = f (ω) f (ω Ω) R 0 00 (ω)γ 00 000 A 000 (ω Ω)γ 0 xx ~ π k k k k V k 2 − − G G − X ~ <,σσ + k (ω) . (6.64) σ mG CHAPTER 6. PARAMAGNETO CONDUCTANCE 150

a) b) ! t 0 ! ⌦ Figure 6.15: The two zeroth order terms. Diagram a) is the bare current-current bubble, and diagram b) is the diamagnetic term associated with the bare bubble.

This is the expression for the (a) diagram depicted in Fig. 6.15, where the ﬁrst term is the paramagnetic component, and the second is the divergent dc component that balances oﬀ the diamagnetic term. The (b) diagram in Fig. 6.15 is the diamagnetic component. From Eq. (6.45) we can see that the diamagnetic component of the unperturbed fermion loop is given by

Z ι X dω ρ (0) = <,σσ(ω) . (6.65) − π Gk h i V kσ 2

From this expression, one can see immediately from Eq. (6.12) that the diamagnetic portion cancels against the second term in Eq. (6.64). Computing the remaining term in Eq. (6.64) and utilizing Eq. (6.12) we ﬁnd σ σ(0)(Ω) = D , (6.66) 1 ιΩτ − 2 2 where we have introduced the Drude conductivity σD = 2e N0D, and D = vFτ/2 is the 2d diﬀusion constant.

The correction

Armed with Eq. (6.12), Eq. (6.58), and Eq. (6.61) we are now able to produce the ﬁnal result for the modiﬁcation to the ac conductivity given by the exchange diagrams. The diamagnetic contributions CHAPTER 6. PARAMAGNETO CONDUCTANCE 151

have been accounted for and removed, so that Eq. (6.12) simpliﬁes to

1 δσ(Ω) = δΠR,(3q)(Ω) + δΠR,(3)(Ω) . −ιΩ xx xx

Combining the terms, we present the modification to the conductance considering the presence of interfacial Rashba spin-orbit coupling in a disordered metallic thin film affixed to a quantum paramagnet lacking long-range magnetic order. A general statement we can make that aids us in n o n o n o n o our presentation is that Re χ ( y) = Re χ (y) and Im χ ( y) = Im χ (y) . Thus, χA ( y) = Q − Q Q − − Q Q − R A χQ(y). Negating y in each of the χ terms allows for combination of the retarded and advanced portions such that:

2 2 Z 2 Z δσ(w) ιπ ν N0 d Q dy F(y) F(y w) + F( y) F( y + w) = J 0 2 − − − − − σ0 2~ (2π) 2π w    xxR yyR zzR xxR zzR   χQ (y) + χQ (y) + χQ (y) χQ (y) + χQ (y)  Q2 + 4Z2  , (6.67)  2 2  ×  1 Q2 ιy 1 Q2 ι (y + w) 1 Q2 ιy 1 Q2 ι (y + w)  2 − 2 − 2 − 2 −

where we have transitioned to unitless integrals, written Ωτ w, and presented with respect to ≡ 2e2 the quantum of conductanct σ0 = h . We have focused on the diagrams for which corrections exist both with and without spin-orbit coupling in order to compare similar terms. The total ac conductivity to Z2 order given by exchange diagrams with maximal ladders is therefore given by σ(w) = σ(0)(w) + δσ(w).

The dc limit

The dc limit is found by taking the external frequency w 0. In shifting to the dc limit we may → write:

y y F(y) F(y w) F( y + w) + F( y) d y s sinh s lim − − − − − = y coth = − y H(y, s) , (6.68) w 0 w dy s 1 cosh ≡ → − s CHAPTER 6. PARAMAGNETO CONDUCTANCE 152

kBTτ where s = ~ is the unitless temperature, and recall that we calculate in the diﬀusive regime such that s 1. Thus, we ﬁnd for the dc limit at non-zero temperatures the conductivity becomes

 yyR 2 2 Z 2 Z  xxR zzR δσ(s) ιπ ν N0 d Q dy  χQ (y) + χQ (y) + χQ (y) = J 0 ,  2 2 H(y s) Q 3 σ0 2~ (2π) 2π  1 2 2 Q ιy −  xxR zzR  χQ (y) + χQ (y) + 4Z2  . (6.69) 3  1 Q2 ιy  2 −

This is the main result we will analyze in the next section.

6.8 Discussion and conclusions

Equation (6.69) is a general result that characterizes the temperature dependent dc modification to the metallic conductivity that arises due to the presence of an affixed PI. The temperature dependence is completely contained in the function H(y, s), where temperature smearing serves to 3 restrict phase space for the frequency integral. The Q and y dependent envelope, 1 Q2 ιy − , 2 − results from three orders of the diffuson pole and is strongly peaked for Q, y 0. This envelope ≈ governs the conversion of spin-to-charge fluctuations across the interface — a process which occurs even in the absence of spin-orbit coupling due to electron scattering against interfacial spin fluctuations causing energy and momentum relaxation, and characterizes an effective spin-Hall angle-type quantity for noise conversion in the N-PI bilayer system we have examined. The spin dynamics of the affixed PI are contained in the spin-susceptibility, χ, and in what comes we will

R xxR yyR zzR assume that the PI is SU(2) symmetric such that χQ(y) = χQ (y) = χQ (y) = χQ (y), and similarly for the advanced component. We first make some structural comments about this result, and subsequently apply our general theory to two types of spin systems. In the first case, the PI spin susceptibility is smooth and effectively constant near Q, y 0, and in the second we examine a ≈ gapless QSL possessing a spinon Fermi surface. CHAPTER 6. PARAMAGNETO CONDUCTANCE 153

In the absence of spin-orbit coupling the spin-fermion interactions — Eq. (6.60) — are traceless. This has immediate implications, namely that the Hartree contributions to the conductivity and quasiparticle scattering rates vanish. The symmetry of the spin-fermion interaction vertex requires that any particle-hole bubbles containing only a single interaction — see Figs. 6.5 and 6.6 — vanish identically due to the trace over spin indices. This finding is in line with past results [206], and we therefore focus on the exchange contributions in the cases with and without spin-orbit coupling. However, including spin-orbit interactions opens the Hartree channels which may result in additional corrections to the contributions we present here at very low temperatures. Our results therefore represent a minimum contribution to the conductivity corrections in the spin-orbit coupling cases in general. Additionally, we mention a peculiar feature of the result in Eq. (6.69) prior to the assumption of SU(2) symmetry. That is, the χyy portion of the spin susceptibility does not enter the spin- orbit coupling term. Spin-rotational symmetry remains unbroken in the term without spin-orbit coupling, and thus the spin susceptibility enters symmetrically in that term. However, the interface breaks inversion symmetry along thez ˆ direction, and we measure conductivity in thex ˆ direction; thus, Rashba spin-orbit coupling gives rise to an effective magnetic field that causes electron spins to precess about they ˆ axis. As a result, no contribution from χyy enters the term proportional to Z2. As mentioned above, corrections to the conductivity depends expressly upon the spin susceptibility of the PI layer, and thus act as a probe of the PI magnetization dynamics. The retarded susceptibility can be written in terms of explicit real and imaginary components, such that

R χQ(y) = RQ(y) + ιIQ(y), and we recall that the real component is even with respect to y while the imaginary component is odd. Because we assume SU(2) symmetry, we can therefore write the correction in terms of the explicit real and imaginary components as:

  2 2 Z 2 Z   δσ(s) ιπ ν N0 d Q dy  RQ(y) + ιIQ(y) RQ(y) + ιIQ(y) = J 0 H(y, s) 3Q2 + 8Z2  . (6.70) ~ 2  3 3  σ0 2 (2π) 2π  1 Q2 ιy 1 Q2 ιy  2 − 2 − CHAPTER 6. PARAMAGNETO CONDUCTANCE 154

Let us now assume that the susceptibility is smooth near Q, y 0 with no strong weights away ≈ from the origin, such that R (y) R and I (y) I y. Under this assumption, we can solve the Q Q ≈ 0 q ≈ 1 integrals without issue and the result to lowest order in y is

2 2 Z δσ(s) ν0N0 2 ∞ dy H(y, s) = J 3R0 4Z I1 . (6.71) σ0 − 2~ − 0 2π y

The leading contribution to the conductivity from the y integral is then

2 2 δσ(s) ν0N0 2 J 3R0 4Z I1 ln (s) , (6.72) σ0 ≈ 2~ − and we recall that s 1. Thus, the contributions both from the term lacking spin-orbit coupling and the term arising due to spin-orbit coupling give logarithmic corrections. Here, each component may be either metallic or insulating depending on the spin structure, and in particular the signs of

R0 and I1. However, the contribution emerging due to spin-orbit coupling is suppressed by a factor of Z2, with Z 1. In the case of corrections to the conductivity arising due to screened electron interactions originally studied by Aronov and Altshuler a universal insulating correction that goes as ln(s) arises due to the exchange diagrams [198; 199]; we see here that non-universal corrections arise at the same order when electron interactions are instead mediated by spin fluctuations if the affixed PI possesses a smooth susceptibility near Q, y 0. ≈ We may instead interface a PI with more complicated spin structure in the bilayer. Consider, for instance, the case of a QSL characterized by a parton description with gapless fermions and a spinon Fermi surface in the large mean-field limit as addressed in Chapter 5. In that case, the N susceptibility is simply a polarization bubble resulting in a Lindhard equation:

Z ι X dω h i χ () = GR (ω + )GK() + GK (ω + )GA(ω) , (6.73) q π q+k q q+k k −4V k 2

where the Green functions depicted are those of the fermionic spinons hosted by the QSL. Solving CHAPTER 6. PARAMAGNETO CONDUCTANCE 155

this is straightforward, and one ﬁnds that the real component is

˜ N0~ h 2i RQ(y) = A+ A 2(κQ) , (6.74) 4(κQ)2 − − −

where N˜ 0 is the spinon density of states at its Fermi level, k˜F, and

q A = θ( z κQ) sgn (z ) z2 (κQ)2 , (6.75) ± | ±| − ± ± −

1 2 − the z = κy (κQ) , and κ = 2k˜Fl . Note that as expected the real component is even as a ± ± function of y. The imaginary component gives

N˜ 0~ IQ(y) = (B+ B ) , (6.76) −4(κQ)2 − −

where q B = θ(κQ z ) (κQ)2 z2 . (6.77) ± − | ±| − ±

Thus we see that the imaginary component is an odd function of y. A point that requires explicit mentioning is that the solution to Eq. (6.73) is calculated at zero temperature. At low temperatures — i.e., where the QSL state survives — temperature enters above the 0T result in powers of T , T˜F where T˜F is the spinon Fermi temperature, however, and thus the ﬁnite T terms are subleading to the 0T term. The envelope function restricts most of the contribution to the region Q, y 1. We then note that y Q2, such that the real and imaginary components of the susceptibility can be expressed as   ˜ ~ N0  y θ( y Q)  RQ(y) | p| | | − 1 , (6.78) ≈ 2  y2 Q2 −  − N˜ 0~ yθ(Q y) IQ(y) p − . (6.79) ≈ − 2 Q2 y2 − CHAPTER 6. PARAMAGNETO CONDUCTANCE 156

Inserting these results into Eq. (6.70) and completing the Q integral to lowest order in y results in

2 2 Z 2 ! δσ(s) 3 ν N0N˜ 0 ∞ dy 1 π Z J 0 , . H(y s) 3/2 (6.80) σ0 ≈ 4 0 2π y − 2 y

We may then complete the y integral to ﬁnd:

2 2 2 2 ! δσ(s) 3 ν N0N˜ 0 0.352π Z J 0 ln (s) + , (6.81) σ0 ≈ − 8π √s where because s 1 we can see that the ln(s) term that arises in the absence of spin-orbit coupling results in a metallic contribution while the term proportional to Z2 is insulating. However, unlike the previous case, here each component contributes at a diﬀerent order in s. We may therefore predict the crossover temperature T ∗ at which the spin-orbit coupling term begins to dominate and the correction becomes insulating. In bilayer systems interfacing metals with insulators, e.g., LaAlO3 aﬃxed to SrTiO3 [207], Rashba spin-orbit coupling parameters have been found to be on the order

~2λ 3 10− eVnm. If we assume a similar coupling strength in a bilayer of the type we consider m ∼ here, then the crossover temperature T ∗ 25mK, which is within reach of modern experimental ≈ techniques. At low temperatures spin fluctuations have been shown to incite an enhancing correction to metallic conductivity in a metal near a ferromagnetic quantum critical point that goes as ln(s) until the critical point and (ln(s))2 at the critical point [200]. Here, we show that spin fluctuations from a coupled QSL can also drive an enhancement to the electrical conductivity of a weakly spin-orbit coupled metallic thin film that goes as ln(s). The condition s 1 is equivalent to the 14 temperature region T 760K (for τ 10− s, a reasonable assumption for most metals), which ∼ may be beyond the stable temperature region of a QSL phase. The upper bound on temperature therefore comes from the QSL itself, not the s 1 condition, and a metal affixed to a gapless QSL material should therefore see a metallic correction to its conductivity throughout the temperature range of a stable QSL phase down to T ∗ 25mK, at which point the insulating correction driven ≈ CHAPTER 6. PARAMAGNETO CONDUCTANCE 157 by interfacial spin-orbit coupling dominates and the Fermi liquid description of the metal layer may fail. In conclusion, we have presented a microscopic calculation of the electrical conductivity at the interface between a weakly spin-orbit coupled metallic thin film and a PI. The metal contains spin-inert disorder, and Rashba spin-orbit coupling arises due to inversion symmetry breaking at the interface. The interplay between disorder, interfacial spin-orbit coupling, and spin fluctuations resulting from the presence of the PI result in corrections to the dc electrical conductivity in the metal. Energy and momentum relaxation due to metallic disorder give rise to diffuson poles strongly peaked near zero, resulting in a probe of the low energy PI spin dynamics. For instance, if the PI susceptibility is smooth and well-behaved near the origin, then we predict a logarithmic correction to the conductivity — that may be either metallic or insulating depending on the spin structure of the PI — resulting from electron-electron interactions mediated by spin fluctuations. If the PI is instead a gapless QSL material with a spinon Fermi surface, an initial logarithmic metallic correction dominates until a critical temperature T ∗ 25mK at which point an insulating ≈ correction arising due to the presence of interfacial Rashba spin-orbit coupling and diverging as

1/2 T − dominates. Corrections to the conductivity in this type of setup therefore contain information about the magnetization dynamics of the insulating layer, and can be utilized as a probe of the aﬃxed PI. In the future it would be interesting to expand this analysis to the case of gapped QSL models. Gapless QSLs have appreciable weight in their susceptibilities near the origin. A gapped QSL, by contrast, is characterized by exponentially suppressed spin dynamics near zero energy, and thus should not generate AA corrections of the type discussed here. A theoretical approach capable of probing energy and momentum away from the origin would therefore be required; perhaps an ac probe could accommodate this requirement, similar to that proposed in, e.g, Ref. [51]. Extending the calculation to the case of magnetic disorder in the metal may also prove enlightening, where it would be possible to compare the strength of corrections arising due to Rashba and Dresselhaus CHAPTER 6. PARAMAGNETO CONDUCTANCE 158 eﬀects. Finally, the calculation here has assumed weak interfacial spin-orbit interactions such that Z 1; expanding the calculation to include strong Rashba spin-orbit interactions would also be of interest. Chapter 7

Conclusion

The work in this thesis is focused on utilizing techniques from spintronics to study unconventional quantum magnets hosting quantum spin liquid ground states in both 1d and 2d. Spin transport phenomena have figured prominently throughout, and in particular we have seen that spin fluctuations are a viable tool with which to probe unconventional quantum magnets. In Chapter 3 we develop a theory supporting the spin Fano factor as a differentiating quantity between spin shot noise generated at a coupling between 1d spin chains and 3d conventional magnets. We show that in the gapless phase, coupled 1d spin chains are subject to Pauli blocking not seen in magnon hosting systems coupled in the same geometry. We then propose an experimental technique for measuring spin shot noise directly that does not rely upon destructive mechanisms to convert spin signals to charge signals in Chapter 4. This experimental technique utilizes a resonator cavity coupled to a transmission line, and we show that simple photon counting at low temperatures can access the dc spin current shot noise. In Chapter 5, we shift focus from quantum magnets in 1d to 2d quantum spin liquids and develop a theory utilizing equilibrium spin fluctuations as a probe of the low energy density of states of various spin liquid models. We demonstrate that quantities of interest such as gap energies and renormalizations due to the presence of gauge fluctuations can be extracted in an equilibrium, table-top experiment utilizing techniques already common in spintron-

159 CHAPTER 7. CONCLUSION 160

ics. Lastly, in Chapter 6 we demonstrate that spin ﬂuctuations across interfaces between metals and unconventional paramagnets result in characteristic magnetoresistive corrections to the metallic conductivity, expanding upon prior work examining Coulomb interactions and electron-electron interactions mediated by spin ﬂuctuations in a monolayer.

Outlook

One major outstanding issue in the search for an uncontroversial instantiation of a quantum spin liquid in 2d is the unambiguous detection of mobile spinons. In Chapter 3 we noted the fact that the system of coupled 1d spin chains we proposed is formally identical to a fractional quantum Hall system wherein two coupled edge modes exchange electrons. The exchange coupling is through vacuum and passes spin ~, in the same way the edge to edge coupling passes an electron in the equivalent fractional quantum Hall case. Fractional quantum Hall experiments have been performed in which edge modes instead couple through the bulk and thus couplings can occur in which fractions of electron charges tunnel between the edges. Developing a spin version of this second case, wherein spin tunneling occurs through the bulk of a quantum spin liquid, would be of extreme interest. In the charge scenario, shot noise measurements have revealed the fractionalized nature of the bulk quasiparticles — in the same vein, it would be interesting to discover if spin shot noise would do the same in the spin case and thus uncover a sign of mobile spinons. A second avenue for future research is including Dzyaloshinskii-Moriya (DM) interactions and impurities in the spin liquid models utilized in Chapter 5. In the HKLM introduced in Chapter 5, for example, we estimated the gap energy at ∆ 0.05J following numerical studies on the ≈ compound herbertsmithite [163]. However, the DM coupling has been estimated at 0.1J in ∼ copper oxides, so that the DM coupling may exceed the spinon gap energy. DM interactions break spin rotational symmetry, and so they may serve to reduce quantum ﬂuctuations and favor phases other than the spin liquid ground state. As a result, their eﬀects on the voltage noise spectrum may CHAPTER 7. CONCLUSION 161

be of interest when attempting to detect the spin liquid ground state of materials thought to be well characterized by the HKLM. In the same vein, impurities may play an important role in the low energy behavior of the compound herbertsmithite. As we have seen, herbertsmithite is thought to host a quantum spin liquid ground state; however, random substitution of magnetic Cu2+ ions at transition metal sites in the lattice may contribute significantly to the spin Hall noise in the low temperature regime. It would therefore be interesting to examine how the spin Hall noise studied in Chapter 5 is renormalized in the presence of magnetic impurity spins in the case of the HKLM. A final potential avenue for future research worth mentioning here is expanding the temperature range of the Kitaev spin liquid state in the system that we have examined. The Kitaev model has two cross-over temperatures proportional to the bond interaction strength K, T K/100, and L ∼ T K. We performed our calculation in the zero temperature limit in order to examine the flux- H ∼ free sector, T < TL. However, in the temperature range TL < T < TH mobile Majorana spinons are thought to exist atop a thermally disordered flux background [157]. These flux excitations may act as a momentum sink for mobile spinons and thus enhance the spin Hall noise generated in the configuration we have proposed. Future research directions are naturally unlimited. We have posed here some of the immediate next steps with respect to the work done in this thesis one might take. This thesis has attempted to demonstrate that spin fluctuations can provide a useful probe of unconventional quantum magnets, and in particular those hosting quantum spin liquid ground states. It is our hope that some of the work contained herein will obtain experimental notice, and we believe we have contributed to a stronger understanding of exotic quantum magnets. Bibliography

[1] R. L. Dragomirova, L. P. Zarbo,ˆ and B. K. Nikolic.´ Spin and charge shot noise in spin hall systems. Europhys. Lett., 101:37004, 2008. [2] Philipp Gegenwart and Simon Trebst. Kitaev matter. Nature Physics, 11(6):444–445, 2015. [3] Shubhayu Chatterjee and Subir Sachdev. Probing excitations in insulators via injection of spin currents. Phys. Rev. B, 92:165113, Oct 2015. [4] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner. Dynamics of a two-dimensional quantum spin liquid: Signatures of emergent majorana fermions and fluxes. Phys. Rev. Lett., 112:207203, May 2014. [5] Leon Balents. Spin liquids in frustrated magnets. Nature, 464(7286):199–208, 03 2010. [6] Lucile Savary and Leon Balents. Quantum spin liquids: a review. Rep. Prog. Phys., 80(016502), November 2016. [7] Subir Sachdev. Kagome-´ and triangular-lattice heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons. Phys. Rev. B, 45:12377–12396, Jun 1992. [8] Sung-Sik Lee and Patrick A. Lee. U(1) gauge theory of the hubbard model: Spin liquid states and possible application to κ-(BEDT-TTF)2cu2(CN)3. Phys. Rev. Lett., 95:036403, Jul 2005. [9] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2 – 111, 2006. [10] L. Faddeev and L. Takhtajan. What is the spin of a spin wave? Physics Letters A, 85(6- 7):375–377, July 1981. [11] Thierry Giamarchi. Quantum Physics in One Dimension. Oxford University Press, Oxford, 2004. [12] Martin Klanjsek.ˇ A critical test of quantum criticality. Physics, 7:74, July 2014. [13] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil. Quantum spin liquids. Science, 367(6475), 2020. 162 BIBLIOGRAPHY 163

[14] M. R. Norman. Colloquium: Herbertsmithite and the search for the quantum spin liquid. Rev. Mod. Phys., 88:041002, Dec 2016.

[15] B J Powell and Ross H McKenzie. Quantum frustration in organic mott insulators: from spin liquids to unconventional superconductors. Rep. Prog. Phys., 74(5):056501, apr 2011.

[16] Hidenori Takagi, Tomohiro Takayama, George Jackeli, Giniyat Khaliullin, and Stephen E. Nagler. Concept and realization of kitaev quantum spin liquids. Nature Reviews Physics, 1(4):264–280, 2019.

[17] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh. Transmission of electrical signals by spin-wave interconversion in a magnetic insulator. Nature, 464:262–266, 2010.

[18] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees. Long-distance transport of magnon spin information in a magnetic insulator at room temperature. Nature Phys., 11(12):1022–1026, 12 2015.

[19] Daichi Hirobe, Masahiro Sato, Takayuki Kawamata, Yuki Shiomi, Ken-ichi Uchida, Ryo Iguchi, Yoji Koike, Sadamichi Maekawa, and Eiji Saitoh. One-dimensional spinon spin currents. Nature Phys., 13(1):30–34, 01 2017.

[20] Tetsuya Minakawa, Yuta Murakami, Akihisa Koga, and Joji Nasu. Majorana-mediated spin transport in kitaev quantum spin liquids. Phys. Rev. Lett., 125:047204, Jul 2020.

[21] X. Jehl, M. Sanquer, R. Calemczuk, and D. Mailly. Detection of doubled shot noise in short normal-metal/ superconductor junctions. Nature, 405(6782):50–53, 2000.

[22] R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu. Direct observation of a fractional charge. Nature, 389(6647):162–164, 1997.

[23] Jairo Sinova, Sergio O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth. Spin hall eﬀects. Rev. Mod. Phys., 87:1213–1260, Oct 2015.

[24] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara. Conversion of spin current into charge current at room temperature: Inverse spin-hall eﬀect. Appl. Phys. Lett., 88(18):182509, 2006.

[25] C. W. Sandweg, Y.Kajiwara, A. V.Chumak, A. A. Serga, V.I. Vasyuchka, M. B. Jungﬂeisch, E. Saitoh, and B. Hillebrands. Spin pumping by parametrically excited exchange magnons. Phys. Rev. Lett., 106(21):216601, May 2011.

[26] V. P. Amin and M. D. Stiles. Spin transport at interfaces with spin-orbit coupling: Formal- ism. Phys. Rev. B, 94:104419, Sep 2016. BIBLIOGRAPHY 164

[27] V. P. Amin and M. D. Stiles. Spin transport at interfaces with spin-orbit coupling: Phe- nomenology. Phys. Rev. B, 94:104420, Sep 2016.

[28] V. P. Amin, P. M. Haney, and M. D. Stiles. Interfacial spin–orbit torques. Journal of Applied Physics, 128(15):151101, 2020.

[29] Mathias Weiler, Matthias Althammer, Franz D. Czeschka, Hans Huebl, Martin S. Wagner, Matthias Opel, Inga-Mareen Imort, Gunter¨ Reiss, Andy Thomas, Rudolf Gross, and Se- bastian T.B. Goennenwein. Local charge and spin currents in magnetothermal landscapes. Phys. Rev. Lett., 108:106602, 2012.

[30] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien. Transport magnetic proximity eﬀects in platinum. Phys. Rev. Lett., 109:107204, Sep 2012.

[31] Yan-Ting Chen, Saburo Takahashi, Hiroyasu Nakayama, Matthias Althammer, Sebastian T. B. Goennenwein, Eiji Saitoh, and Gerrit E. W. Bauer. Theory of spin hall magnetoresistance. Phys. Rev. B, 87:144411, Apr 2013.

[32] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprags,¨ M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh. Spin hall magnetoresistance induced by a nonequilibrium proximity eﬀect. Phys. Rev. Lett., 110:206601, May 2013.

[33] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef. Comparative measurements of inverse spin hall eﬀects and magnetoresistance in yig/pt and yig/ta. Phys. Rev. B, 87:174417, May 2013.

[34] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef. Spin-hall magnetoresistance in platinum on yttrium iron garnet: Dependence on platinum thickness and in-plane/out-of-plane magnetization. Phys. Rev. B, 87:184421, May 2013.

[35] Matthias Althammer, Sibylle Meyer, Hiroyasu Nakayama, Michael Schreier, Stephan Alt- mannshofer, Mathias Weiler, Hans Huebl, Stephan Geprags,¨ Matthias Opel, Rudolf Gross, Daniel Meier, Christoph Klewe, Timo Kuschel, Jan-Michael Schmalhorst, Gunter¨ Reiss, Liming Shen, Arunava Gupta, Yan-Ting Chen, Gerrit E. W. Bauer, Eiji Saitoh, and Sebastian T. B. Goennenwein. Quantitative study of the spin hall magnetoresistance in ferromagnetic insulator/normal metal hybrids. Phys. Rev. B, 87:224401, Jun 2013.

[36] Xian-Peng Zhang, F. Sebastian Bergeret, and Vitaly N. Golovach. Theory of spin hall magnetoresistance from a microscopic perspective. Nano Letters, 19(9):6330–6337, 09 2019.

[37] J. H. Han, C. Song, F. Li, Y. Y. Wang, G. Y. Wang, Q. H. Yang, and F. Pan. Antiferromagnet- controlled spin current transport in srmno3/Pt hybrids. Phys. Rev. B, 90:144431, Oct 2014. BIBLIOGRAPHY 165

[38] A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda, B. J. van Wees, and T. T. M. Palstra. Spin-hall magnetoresistance and spin seebeck eﬀect in spin-spiral and paramagnetic phases of multiferroic cocr2o4 ﬁlms. Phys. Rev. B, 92:224410, Dec 2015. [39] Geert R. Hoogeboom, Aisha Aqeel, Timo Kuschel, Thomas T. M. Palstra, and Bart J. van Wees. Negative spin hall magnetoresistance of pt on the bulk easy-plane antiferromagnet nio. Applied Physics Letters, 111(5):052409, 2017.

[40] Dazhi Hou, Zhiyong Qiu, Joseph Barker, Koji Sato, Kei Yamamoto, Saul¨ Velez,´ Juan M. Gomez-Perez, Luis E. Hueso, Felix` Casanova, and Eiji Saitoh. Tunable sign change of spin hall magnetoresistance in Pt/NiO/YIG structures. Phys. Rev. Lett., 118:147202, Apr 2017.

[41] A Aqeel, M Mostovoy, B J van Wees, and T T M Palstra. Spin-hall magnetoresistance in multidomain helical spiral systems. Journal of Physics D: Applied Physics, 50(17):174006, mar 2017.

[42] Johanna Fischer, Olena Gomonay, Richard Schlitz, Kathrin Ganzhorn, Nynke Vlietstra, Matthias Althammer, Hans Huebl, Matthias Opel, Rudolf Gross, Sebastian T. B. Goennen- wein, and Stephan Geprags.¨ Spin hall magnetoresistance in antiferromagnet/heavy-metal heterostructures. Phys. Rev. B, 97:014417, Jan 2018.

[43] Juan M. Gomez-Perez, Xian-Peng Zhang, Francesco Calavalle, Maxim Ilyn, Carmen Gonzalez-Orellana,´ Marco Gobbi, Celia Rogero, Andrey Chuvilin, Vitaly N. Golovach, Luis E. Hueso, F. Sebastian Bergeret, and Felix` Casanova. Strong interfacial exchange ﬁeld in a heavy metal/ferromagnetic insulator system determined by spin hall magnetoresistance. Nano Letters, 20(9):6815–6823, 09 2020.

[44] L. Baldrati, A. Ross, T. Niizeki, C. Schneider, R. Ramos, J. Cramer, O. Gomonay, M. Filian- ina, T. Savchenko, D. Heinze, A. Kleibert, E. Saitoh, J. Sinova, and M. Klaui.¨ Full angular dependence of the spin hall and ordinary magnetoresistance in epitaxial antiferromagnetic nio(001)/pt thin ﬁlms. Phys. Rev. B, 98:024422, Jul 2018.

[45] Yang Cheng, Sisheng Yu, Adam S. Ahmed, Menglin Zhu, You Rao, Maryam Ghazisaeidi, Jinwoo Hwang, and Fengyuan Yang. Anisotropic magnetoresistance and nontrivial spin hall magnetoresistance in Pt/α Fe o bilayers. Phys. Rev. B, 100:220408, Dec 2019. − 2 3 [46] L. Liang, J. Shan, Q. H. Chen, J. M. Lu, G. R. Blake, T. T. M. Palstra, G. E. W. Bauer, B. J. van Wees, and J. T. Ye. Gate-controlled magnetoresistance of a paramagnetic- insulator—platinum interface. Phys. Rev. B, 98:134402, Oct 2018.

[47] Michaela Lammel, Richard Schlitz, Kevin Geishendorf, Denys Makarov, Tobias Kosub, Savio Fabretti, Helena Reichlova, Rene Huebner, Kornelius Nielsch, Andy Thomas, and Sebastian T. B. Goennenwein. Spin hall magnetoresistance in heterostructures consisting of noncrystalline paramagnetic yig and pt. Applied Physics Letters, 114(25):252402, 2019. BIBLIOGRAPHY 166

[48] Koichi Oyanagi, Juan M. Gomez-Perez, Xian-Peng Zhang, Takashi Kikkawa, Yao Chen, Edurne Sagasta, Luis E. Hueso, Vitaly N. Golovach, F. Sebastian Bergeret, Felix` Casanova, and Eiji Saitoh. Paramagnetic spin hall magnetoresistance. 2020.

[49] Joshua Aftergood and So Takei. Noise in tunneling spin current across coupled quantum spin chains. Phys. Rev. B, 97:014427, Jan 2018.

[50] Joshua Aftergood, Mircea Trif, and So Takei. Detecting spin current noise in quantum magnets with photons. Phys. Rev. B, 99:174422, May 2019.

[51] Joshua Aftergood and So Takei. Probing quantum spin liquids in equilibrium using the inverse spin hall eﬀect. Phys. Rev. Research, 2:033439, Sep 2020.

[52] Yaroslav Tserkovnyak and Arne Brataas. Shot noise in ferromagnet–normal metal systems. Phys. Rev. B, 64(21):214402, 2001.

[53] E. G. Mishchenko and B. I. Halperin. Transport equations for a two-dimensional electron gas with spin-orbit interaction. Phys. Rev. B, 68(4):045317, 2003.

[54] A. Lamacraft. Shot noise of spin-polarized electrons. Phys. Rev. B, 69:081301, Feb 2004.

[55] Wolfgang Belzig and Malek Zareyan. Spin-ﬂip noise in a multiterminal spin valve. Phys. Rev. B, 69:140407, April 2004.

[56] Jørn Foros, Arne Brataas, Yaroslav Tserkovnyak, and Gerrit E. W. Bauer. Magnetization noise in magnetoelectronic nanostructures. Phys. Rev. Lett., 95(1):016601, 2005.

[57] M. Zareyan and W. Belzig. Shot noise of spin current in ferromagnet-normal-metal systems. EPL (Europhysics Letters), 70(6):817, 2005.

[58] A. L. Chudnovskiy, J. Swiebodzinski, and A. Kamenev. Spin-torque shot noise in magnetic tunnel junctions. Phys. Rev. Lett., 101(6):066601, 2008.

[59] Jonathan Meair, Peter Stano, and Philippe Jacquod. Measuring spin accumulations with current noise. Phys. Rev. B, 84:073302, Aug 2011.

[60] Tomonori Arakawa, Junichi Shiogai, Mariusz Ciorga, Martin Utz, Dieter Schuh, Makoto Kohda, Junsaku Nitta, Dominique Bougeard, Dieter Weiss, Teruo Ono, and Kensuke Kobayashi. Shot noise induced by nonequilibrium spin accumulation. Phys. Rev. Lett., 114:016601, Jan 2015.

[61] Baigeng Wang, Jian Wang, and Hong Guo. Shot noise of spin current. Phys. Rev. B, 69:153301, Apr 2004.

[62] O. Sauret and D. Feinberg. Spin-current shot noise as a probe of interactions in mesoscopic systems. Phys. Rev. Lett., 92:106601, Mar 2004. BIBLIOGRAPHY 167

[63] Akashdeep Kamra, Friedrich P. Witek, Sibylle Meyer, Hans Huebl, Stephan Geprags,¨ Rudolf Gross, Gerrit E. W. Bauer, and Sebastian T. B. Goennenwein. Spin hall noise. Phys. Rev. B, 90:214419, Dec 2014.

[64] Akashdeep Kamra and Wolfgang Belzig. Super-poissonian shot noise of squeezed-magnon mediated spin transport. Phys. Rev. Lett., 116:146601, Apr 2016.

[65] Akashdeep Kamra and Wolfgang Belzig. Spin pumping and shot noise in ferrimagnets: Bridging ferro- and antiferromagnets. Phys. Rev. Lett., 119:197201, Nov 2017.

[66] M. Matsuo, Y. Ohnuma, T. Kato, and S. Maekawa. Spin current noise of the spin Seebeck eﬀect and spin pumping. ArXiv e-prints, November 2017.

[67] Sebastian T. B. Goennenwein, Richard Schlitz, Matthias Pernpeintner, Kathrin Ganzhorn, Matthias Althammer, Rudolf Gross, and Hans Huebl. Non-local magnetoresistance in yig/pt nanostructures. Appl. Phys. Lett., 107(17):172405, 2015.

[68] Junxue Li, Yadong Xu, Mohammed Aldosary, Chi Tang, Zhisheng Lin, Shufeng Zhang, Roger Lake, and Jing Shi. Observation of magnon-mediated current drag in pt/yttrium iron garnet/pt(ta) trilayers. Nature Comm., 7(10858), 03 2016.

[69] P. Jordan and E. Wigner. Uber¨ das paulische aquivalenzverbot.¨ Zeitschrift f¨urPhysik, 47(9):631–651, 1928.

[70] Andreas Klumper.¨ Integrability of quantum chains: Theory and applications to the spin-1/2 XXZ chain, pages 349–379. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004.

[71] Hans-Jurgen¨ Mikeska and Alexei K. Kolezhuk. One-dimensional magnetism, pages 1–83. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004.

[72] P. R. Hammar, M. B. Stone, Daniel H. Reich, C. Broholm, P. J. Gibson, M. M. Turnbull, C. P. Landee, and M. Oshikawa. Characterization of a quasi-one-dimensional spin-1/2 magnet which is gapless and paramagnetic for gµ h . j and k t j. Phys. Rev. B, 59:1008–1015, B B Jan 1999.

[73] M. B. Stone, D. H. Reich, C. Broholm, K. Lefmann, C. Rischel, C. P. Landee, and M. M. 1 Turnbull. Extended quantum critical phase in a magnetized spin- 2 antiferromagnetic chain. Phys. Rev. Lett., 91:037205, Jul 2003.

[74] T. Lancaster, S. J. Blundell, M. L. Brooks, P. J. Baker, F. L. Pratt, J. L. Manson, C. P. Landee, and C. Baines. Magnetic order in the quasi-one-dimensional spin-12 molecular chain compound copper pyrazine dinitrate. Phys. Rev. B, 73:020410, Jan 2006.

[75] H. Kuhne,¨ H.-H. Klauss, S. Grossjohann, W. Brenig, F. J. Litterst, A. P. Reyes, P. L. Kuhns, 1 M. M. Turnbull, and C. P. Landee. Quantum critical dynamics of an s = 2 antiferromagnetic heisenberg chain studied by 13C nmr spectroscopy. Phys. Rev. B, 80:045110, Jul 2009. BIBLIOGRAPHY 168

[76] H. Kuhne,¨ A. A. Zvyagin, M. Gunther,¨ A. P. Reyes, P. L. Kuhns, M. M. Turnbull, C. P. Landee, and H.-H. Klauss. Dynamics of a heisenberg spin chain in the quantum critical regime: Nmr experiment versus eﬀective ﬁeld theory. Phys. Rev. B, 83:100407, Mar 2011.

[77] M. Heiblum. Quantum shot noise in edge channels. physica status solidi (b), 243(14):3604– 3616, 2006.

[78] C. L. Kane and Matthew P. A. Fisher. Edge-State Transport, pages 109–159. Wiley-VCH Verlag GmbH, 2007.

[79] S. Maekawa, S.O. Valenzuela, E. Saitoh, and T. Kimura. Spin Current. Series on Semicon- ductor Science and Technology. OUP Oxford, 2012.

[80] Arne Brataas, Andrew D. Kent, and Hideo Ohno. Current-induced torques in magnetic materials. Nat Mater, 11:372–381, May 2012.

[81] Paul M. Haney, Hyun-Woo Lee, Kyung-Jin Lee, Aurelien´ Manchon, and M. D. Stiles. Cur- rent induced torques and interfacial spin-orbit coupling: Semiclassical modeling. Phys. Rev. B, 87:174411, May 2013.

[82] Yaroslav Tserkovnyak and Scott A. Bender. Spin hall phenomenology of magnetic dynamics. Phys. Rev. B, 90:014428, Jul 2014.

[83] Alexander O. Gogolin. Bosonization and Strongly Correlated Systems. Cambridge Univer- sity Press, 1998.

[84] E. Miranda. Introduction to bosonization. Brazilian Journal of Physics, 33:3 – 35, 03 2003.

[85] James D. Johnson, Samuel Krinsky, and Barry M. McCoy. Vertical-arrow correlation length in the eight-vertex model and the low-lying excitations of the x y z hamiltonian. Phys. − − Rev. A, 8:2526–2547, Nov 1973.

[86] J. Rammer. Quantum Field Theory of Non-equilibrium States. Cambridge University Press, 2007.

[87] Alex Kamenev. Field Theory of Non-Equilibrium Systems. Cambridge University Press, Cambridge, 2011.

[88] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees. Magnon spin transport driven by the magnon chemical potential in a magnetic insulator. Phys. Rev. B, 94:014412, Jul 2016.

[89] Juan Shan, Ludo J. Cornelissen, Nynke Vlietstra, Jamal Ben Youssef, Timo Kuschel, Rem- bert A. Duine, and Bart J. van Wees. Inﬂuence of yttrium iron garnet thickness and heater opacity on the nonlocal transport of electrically and thermally excited magnons. Phys. Rev. B, 94:174437, Nov 2016. BIBLIOGRAPHY 169

[90] Andreas Ruckriegel,¨ Peter Kopietz, Dmytro A. Bozhko, Alexander A. Serga, and Burkard Hillebrands. Magnetoelastic modes and lifetime of magnons in thin yttrium iron garnet ﬁlms. Phys. Rev. B, 89:184413, May 2014.

[91] T. Holstein and H. Primakoﬀ. Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev., 58(12):1098–1113, Dec 1940.

[92] C. de C. Chamon and X. G. Wen. Resonant tunneling in the fractional quantum hall regime. Phys. Rev. Lett., 70:2605–2608, Apr 1993.

[93] C. L. Kane and Matthew P. A. Fisher. Nonequilibrium noise and fractional charge in the quantum hall eﬀect. Phys. Rev. Lett., 72(5):724–727, Jan 1994.

[94] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha. Current rectiﬁcation by pauli exclusion in a weakly coupled double quantum dot system. Science, 297:1313–1317, 2002.

[95] So Takei, Bernd Rosenow, and Ady Stern. Noise due to neutral modes in the ν = 2/3 fractional quantum hall state. Phys. Rev. B, 91:241104, Jun 2015.

[96] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann. Detection and quantification of inverse spin hall effect from spin pumping in permalloy/normal metal bilayers. Phys. Rev. B, 82:214403, Dec 2010.

[97] David C. Johnston. Chapter 1 Normal-state magnetic properties of single-layer cuprate high-temperature superconductors and related materials, volume 10 of Handbook of Mag- netic Materials. Elsevier, 1997.

[98] Ya. M. Blanter and M. Buttiker.¨ Shot noise in mesoscopic conductors. Phys. Rep., 336:1– 166, 2000.

[99] Eran Sela, Yuval Oreg, Felix von Oppen, and Jens Koch. Fractional shot noise in the kondo regime. Phys. Rev. Lett., 97:086601, Aug 2006.

[100] C. W. J. Beenakker, C. Emary, M. Kindermann, and J. L. van Velsen. Proposal for production and detection of entangled electron-hole pairs in a degenerate electron gas. Phys. Rev. Lett., 91:147901, Oct 2003.

[101] M. Matsuo, Y. Ohnuma, T. Kato, and S. Maekawa. Spin current noise of the spin seebeck eﬀect and spin pumping. Phys. Rev. Lett., 120:037201, Jan 2018.

[102] E. G. Mishchenko. Shot noise in a diﬀusive ferromagnetic-paramagnetic-ferromagnetic spin valve. Phys. Rev. B, 68:100409, Sep 2003.

[103] Nikolai A Sinitsyn and Yuriy V Pershin. The theory of spin noise spectroscopy: a review. Rep. Prog. Phys., 79(10):106501, 2016. BIBLIOGRAPHY 170

[104] Carmine Granata and Antonio Vettoliere. Nano superconducting quantum interference device: A powerful tool for nanoscale investigations. Physics Reports, 614:1 – 69, 2016. Nano Superconducting Quantum Interference device: a powerful tool for nanoscale investigations.

[105] Toeno van der Sar, Francesco Casola, Ronald Walsworth, and Amir Yacoby. Nanometre- scale probing of spin waves using single electron spins. Nature Communications, 6:7886 EP –, 08 2015.

[106] Francesco Casola, Toeno van der Sar, and Amir Yacoby. Probing condensed matter physics with magnetometry based on nitrogen-vacancy centres in diamond. Nature Reviews Mate- rials, 3:17088 EP –, 01 2018.

[107] B. Flebus and Y. Tserkovnyak. Quantum-impurity relaxometry of magnetization dynamics. Phys. Rev. Lett., 121:187204, Nov 2018.

[108] M. H. Devoret, D. Esteve, H. Grabert, G.-L. Ingold, H. Pothier, and C. Urbina. Eﬀect of the electromagnetic environment on the coulomb blockade in ultrasmall tunnel junctions. Phys. Rev. Lett., 64(15):1824–1827, Apr 1990.

[109] Gert-Ludwig Ingold and Yu. V. Nazarov. Charge tunneling rates in ultrasmall junctions. In H. Grabert and M. H. Devoret, editors, Single Charge Tunneling, NATO ASI Series B, chapter 2, pages 21–107. Plenum Press, New York, 1992.

[110] A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf. Intro- duction to quantum noise, measurement, and ampliﬁcation. Rev. Mod. Phys., 82(2):1155– 1208, Apr 2010.

[111] A. Crepieux,´ R. Guyon, P. Devillard, and T. Martin. Electron injection in a nanotube: Noise correlations and entanglement. Phys. Rev. B, 67(20):205408, 2003.

[112] J. Sirker. The luttinger liquid and integrable models. International Journal of Modern Physics B, 26(22):1244008, September 2012.

[113] Kunihiro Inomata, Zhirong Lin, Kazuki Koshino, William D. Oliver, Jaw-Shen Tsai, Tsuyoshi Yamamoto, and Yasunobu Nakamura. Single microwave-photon detector using an artiﬁcial -type three-level system. Nature Communications, 7:12303 EP –, 07 2016.

[114] C. W. J. Beenakker and H. Schomerus. Counting statistics of photons produced by electronic shot noise. Phys. Rev. Lett., 86:700–703, Jan 2001.

[115] C. W. J. Beenakker and H. Schomerus. Antibunched photons emitted by a quantum point contact out of equilibrium. Phys. Rev. Lett., 93:096801, Aug 2004.

[116] Alexei Kitaev and John Preskill. Topological entanglement entropy. Phys. Rev. Lett., 96:110404, Mar 2006. BIBLIOGRAPHY 171

[117] Michael Levin and Xiao-Gang Wen. Detecting topological order in a ground state wave function. Phys. Rev. Lett., 96:110405, Mar 2006.

[118] T. Senthil and Matthew P. A. Fisher. Z2 gauge theory of electron fractionalization in strongly correlated systems. Phys. Rev. B, 62:7850–7881, Sep 2000.

[119] T. Senthil and Matthew P. A. Fisher. Fractionalization in the cuprates: Detecting the topological order. Phys. Rev. Lett., 86:292–295, Jan 2001.

[120] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito. Spin liquid state in an organic mott insulator with a triangular lattice. Phys. Rev. Lett., 91:107001, Sep 2003.

[121] A. Olariu, P. Mendels, F. Bert, F. Duc, J. C. Trombe, M. A. de Vries, and A. Harrison. 17O nmr study of the intrinsic magnetic susceptibility and spin dynamics of the quantum kagome antiferromagnet zncu3(OH)6cl2. Phys. Rev. Lett., 100:087202, Feb 2008. [122] Mingxuan Fu, Takashi Imai, Tian-Heng Han, and Young S. Lee. Evidence for a gapped spin- liquid ground state in a kagome heisenberg antiferromagnet. Science, 350(6261):655–658, 2015.

[123] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H. Takagi. A spin–orbital-entangled quantum liquid on a honeycomb lattice. Nature, 554:341 EP –, 02 2018.

[124] F. Bert, S. Nakamae, F. Ladieu, D. L’Hote,ˆ P. Bonville, F. Duc, J.-C. Trombe, and 1 P. Mendels. Low temperature magnetization of the s = 2 kagome antiferromagnet Zncu3(OH)6cl2. Phys. Rev. B, 76:132411, Oct 2007. [125] Minoru Yamashita, Norihito Nakata, Yuichi Kasahara, Takahiko Sasaki, Naoki Yoneyama, Norio Kobayashi, Satoshi Fujimoto, Takasada Shibauchi, and Yuji Matsuda. Thermal- transport measurements in a quantum spin-liquid state of the frustrated triangular magnet -(bedt-ttf)2cu2(cn)3. Nature Physics, 5:44 EP –, 11 2008.

[126] Satoshi Yamashita, Takashi Yamamoto, Yasuhiro Nakazawa, Masafumi Tamura, and Reizo Kato. Gapless spin liquid of an organic triangular compound evidenced by thermodynamic measurements. Nature Comm., 2:275, 04 2011.

[127] Satoshi Yamashita, Yasuhiro Nakazawa, Masaharu Oguni, Yugo Oshima, Hiroyuki Nojiri, Yasuhiro Shimizu, Kazuya Miyagawa, and Kazushi Kanoda. Thermodynamic properties of a spin-1/2 spin-liquid state in a [kappa]-type organic salt. Nature Phys., 4(6):459–462, 06 2008.

[128] Minoru Yamashita, Norihito Nakata, Yoshinori Senshu, Masaki Nagata, Hiroshi M. Ya- mamoto, Reizo Kato, Takasada Shibauchi, and Yuji Matsuda. Highly mobile gapless excitations in a two-dimensional candidate quantum spin liquid. Science, 328(5983):1246–1248, 2010. BIBLIOGRAPHY 172

[129] Daichi Hirobe, Masahiro Sato, Yuki Shiomi, Hidekazu Tanaka, and Eiji Saitoh. Magnetic thermal conductivity far above the neel´ temperature in the kitaev-magnet candidate α rucl . − 3 Phys. Rev. B, 95:241112, Jun 2017.

[130] Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, Sixiao Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, T. Shibauchi, and Y. Matsuda. Majorana quantization and half-integer thermal quantum hall eﬀect in a kitaev spin liquid. Nature, 559(7713):227–231, 2018.

[131] P. Bourgeois-Hope, F. Laliberte,´ E. Lefranc¸ois, G. Grissonnanche, S. Rene´ de Cotret, R. Gordon, S. Kitou, H. Sawa, H. Cui, R. Kato, L. Taillefer, and N. Doiron-Leyraud. Ther- mal conductivity of the quantum spin liquid candidate etme3Sb[Pd(dmit)2]2: No evidence of mobile gapless excitations. Phys. Rev. X, 9:041051, Dec 2019.

[132] J. M. Ni, B. L. Pan, B. Q. Song, Y. Y. Huang, J. Y. Zeng, Y. J. Yu, E. J. Cheng, L. S. Wang, D. Z. Dai, R. Kato, and S. Y. Li. Absence of magnetic thermal conductivity in the quantum spin liquid candidate etme3Sb[Pd(dmit)2]2. Phys. Rev. Lett., 123:247204, Dec 2019. [133] J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M. Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H. Chung, D. G. Nocera, and Y. S. Lee. Spin dynamics of the spin-1/2 kagome lattice antiferromagnet zncu3(OH)6cl2. Phys. Rev. Lett., 98:107204, Mar 2007. [134] M. A. de Vries, J. R. Stewart, P. P. Deen, J. O. Piatek, G. J. Nilsen, H. M. Rønnow, and A. Harrison. Scale-free antiferromagnetic ﬂuctuations in the s = 1/2 kagome antiferromagnet herbertsmithite. Phys. Rev. Lett., 103:237201, Dec 2009.

[135] J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M. Bartlett, Y. Qiu, D. G. Nocera, and 1 Y. S. Lee. Dynamic scaling in the susceptibility of the spin- 2 kagome lattice antiferromagnet herbertsmithite. Phys. Rev. Lett., 104:147201, Apr 2010.

[136] A. Banerjee, C. A. Bridges, J. Q. Yan, A. A. Aczel, L. Li, M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler. Proximate kitaev quantum spin liquid behaviour in a honeycomb magnet. Nature Mater., 15:733 EP –, 04 2016.

[137] T. Yildirim and A. B. Harris. Magnetic structure and spin waves in the kagome´ jarosite compound Kfe3(SO4)2(OH)6. Phys. Rev. B, 73:214446, Jun 2006. [138] Simeng Yan, David A. Huse, and Steven R. White. Spin-liquid ground state of the s = 1/2 kagome heisenberg antiferromagnet. Science, 332(6034):1173–1176, 2011.

[139] Harald O. Jeschke, Francesc Salvat-Pujol, and Roser Valent´ı. First-principles determination 1 of heisenberg hamiltonian parameters for the spin- 2 kagome antiferromagnet zncu3(oh)6cl2. Phys. Rev. B, 88:075106, Aug 2013. BIBLIOGRAPHY 173

[140] Yin-Chen He, Michael P. Zaletel, Masaki Oshikawa, and Frank Pollmann. Signatures of dirac cones in a dmrg study of the kagome heisenberg model. Phys. Rev. X, 7:031020, Jul 2017.

[141] Olexei I. Motrunich. Variational study of triangular lattice spin-12 model with ring ex- changes and spin liquid state in κ-(ET)2cu2(CN)3. Phys. Rev. B, 72:045105, Jul 2005. [142] Cody P. Nave and Patrick A. Lee. Transport properties of a spinon fermi surface coupled to a u(1) gauge ﬁeld. Phys. Rev. B, 76:235124, Dec 2007.

[143] Sung-Sik Lee. Stability of the u(1) spin liquid with a spinon fermi surface in 2 + 1 dimensions. Phys. Rev. B, 78:085129, Aug 2008.

[144] Sung-Sik Lee. Low-energy eﬀective theory of fermi surface coupled with u(1) gauge ﬁeld in 2 + 1 dimensions. Phys. Rev. B, 80:165102, Oct 2009.

[145] Sambuddha Sanyal, Kusum Dhochak, and Subhro Bhattacharjee. Interplay of uniform u(1) quantum spin liquid and magnetic phases in rare-earth pyrochlore magnets: A fermionic parton approach. Phys. Rev. B, 99:134425, Apr 2019.

[146] G. Jackeli and G. Khaliullin. Mott insulators in the strong spin-orbit coupling limit: From heisenberg to a quantum compass and kitaev models. Phys. Rev. Lett., 102:017205, Jan 2009.

[147] Saptarshi Mandal and Naveen Surendran. Exactly solvable kitaev model in three dimensions. Phys. Rev. B, 79:024426, Jan 2009.

[148] Joji Nasu, Masafumi Udagawa, and Yukitoshi Motome. Vaporization of kitaev spin liquids. Phys. Rev. Lett., 113:197205, Nov 2014.

[149] J. Knolle and R. Moessner. A ﬁeld guide to spin liquids. Annual Review of Condensed Matter Physics, 10(1):451–472, 2019.

[150] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh. Observation of the spin seebeck eﬀect. Nature, 455:778–781, 2008.

[151] Kenichi Uchida, Hiroto Adachi, Takeru Ota, Hiroyasu Nakayama, Sadamichi Maekawa, and Eiji Saitoh. Observation of longitudinal spin-seebeck eﬀect in magnetic insulators. Appl. Phys. Lett., 97(17):172505, 2010.

[152] Wei Han, Sadamichi Maekawa, and Xin-Cheng Xie. Spin current as a probe of quantum materials. Nature Mater., 2019.

[153] Herbert B. Callen and Theodore A. Welton. Irreversibility and generalized noise. Phys. Rev., 83:34–40, Jul 1951. BIBLIOGRAPHY 174

[154] R Kubo. The ﬂuctuation-dissipation theorem. Reports on Progress in Physics, 29(1):255– 284, jan 1966.

[155] Akashdeep Kamra and Wolfgang Belzig. Magnon-mediated spin current noise in ferromagnet nonmagnetic conductor hybrids. Phys. Rev. B, 94:014419, Jul 2016. | [156] Serge Florens and Antoine Georges. Slave-rotor mean-ﬁeld theories of strongly correlated systems and the mott transition in ﬁnite dimensions. Phys. Rev. B, 70:035114, Jul 2004.

[157] Joji Nasu, Masafumi Udagawa, and Yukitoshi Motome. Thermal fractionalization of quantum spins in a kitaev model: Temperature-linear speciﬁc heat and coherent transport of majorana fermions. Phys. Rev. B, 92:115122, Sep 2015.

[158] I. Rousochatzakis, S. Kourtis, J. Knolle, R. Moessner, and N. B. Perkins. Quantum spin liquid at ﬁnite temperature: Proximate dynamics and persistent typicality. Phys. Rev. B, 100:045117, Jul 2019.

[159] L. A. Hall. Survey of Electrical Resistivity Measurements on 16 Pure Metals in the Temper- ature Range 0 to 273◦ K. National Bureau of Standards, 1964.

[160] P. D. Desai, T. K. Chu, H. M. James, and C. Y. Ho. Electrical resistivity of selected elements. J. Phys. Chem. Ref. Data, 13(4):1069–1096, 1984.

[161] Darshan G. Joshi, Andreas P. Schnyder, and So Takei. Detecting end states of topological quantum paramagnets via spin hall noise spectroscopy. Phys. Rev. B, 98:064401, Aug 2018.

[162] Fa Wang and Ashvin Vishwanath. Spin-liquid states on the triangular and kagome´ lattices: A projective-symmetry-group analysis of schwinger boson states. Phys. Rev. B, 74:174423, Nov 2006.

[163] Matthias Punk, Debanjan Chowdhury, and Subir Sachdev. Topological excitations and the dynamic structure factor of spin liquids on the kagome lattice. Nature Phys., 10:289 EP –, 03 2014. z 1 [164] Yuan-Ming Lu, Ying Ran, and Patrick A. Lee. 2 spin liquids in the s = 2 heisenberg model on the kagome lattice: A projective symmetry-group study of schwinger fermion mean-ﬁeld states. Phys. Rev. B, 83:224413, Jun 2011.

[165] Yejin Huh, Lars Fritz, and Subir Sachdev. Quantum criticality of the kagome antiferromagnet with dzyaloshinskii-moriya interactions. Phys. Rev. B, 81:144432, Apr 2010.

[166] Yao-Dong Li, Xiaoqun Wang, and Gang Chen. Anisotropic spin model of strong spin-orbit- coupled triangular antiferromagnets. Phys. Rev. B, 94:035107, Jul 2016.

[167] Changle Liu, Rong Yu, and Xiaoqun Wang. Semiclassical ground-state phase diagram and multi-q phase of a spin-orbit-coupled model on triangular lattice. Phys. Rev. B, 94:174424, Nov 2016. BIBLIOGRAPHY 175

[168] Yao-Dong Li, Yuan-Ming Lu, and Gang Chen. Spinon fermi surface u(1) spin liquid in

the spin-orbit-coupled triangular-lattice mott insulator ybmggao4. Phys. Rev. B, 96:054445, Aug 2017.

[169] Patrick A. Lee and Naoto Nagaosa. Gauge theory of the normal state of high-tc superconductors. Phys. Rev. B, 46:5621–5639, Sep 1992.

[170] Joseph Polchinski. Low-energy dynamics of the spinon-gauge system. Nuclear Physics B, 422(3):617 – 633, 1994.

[171] Yong Baek Kim, Akira Furusaki, Xiao-Gang Wen, and Patrick A. Lee. Gauge-invariant response functions of fermions coupled to a gauge ﬁeld. Phys. Rev. B, 50:17917–17932, Dec 1994.

[172] Leon Balents and Oleg A. Starykh. Collective spinon spin wave in a magnetized u(1) spin liquid. Phys. Rev. B, 101:020401, Jan 2020.

[173] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner. Dynamics of fractionalization in quantum spin liquids. Phys. Rev. B, 92:115127, Sep 2015.

[174] M. Hermanns, I. Kimchi, and J. Knolle. Physics of the kitaev model: Fractionalization, dynamic correlations, and material connections. Annual Review of Condensed Matter Physics, 9(1):17–33, 2018.

[175] Johannes Knolle, Subhro Bhattacharjee, and Roderich Moessner. Dynamics of a quantum spin liquid beyond integrability: The kitaev-heisenberg-Γ model in an augmented parton mean-ﬁeld theory. Phys. Rev. B, 97:134432, Apr 2018.

[176] Elliott H. Lieb. Flux phase of the half-ﬁlled band. Phys. Rev. Lett., 73:2158–2161, Oct 1994.

[177] G. Baskaran, Saptarshi Mandal, and R. Shankar. Exact results for spin dynamics and fractionalization in the kitaev model. Phys. Rev. Lett., 98:247201, Jun 2007.

[178] S.O. Kasap. Principles of Electronic Materials and Devices. McGraw-Hill International Edition. McGraw-Hill, 2006.

[179] Tian-Heng Han, Joel S. Helton, Shaoyan Chu, Daniel G. Nocera, Jose A. Rodriguez-Rivera, Collin Broholm, and Young S. Lee. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature, 492(7429):406–410, 12 2012.

[180] Tian-Heng Han, M. R. Norman, J.-J. Wen, Jose A. Rodriguez-Rivera, Joel S. Helton, Collin Broholm, and Young S. Lee. Correlated impurities and intrinsic spin-liquid physics in the kagome material herbertsmithite. Phys. Rev. B, 94:060409, Aug 2016. BIBLIOGRAPHY 176

[181] Danna E. Freedman, Tianheng H. Han, Andrea Prodi, Peter Muller,¨ Qing-Zhen Huang, Yu- Sheng Chen, Samuel M. Webb, Young S. Lee, Tyrel M. McQueen, and Daniel G. Nocera. Site speciﬁc x-ray anomalous dispersion of the geometrically frustrated kagomemagnet,´ herbertsmithite, zncu3(oh)6cl2. Journal of the American Chemical Society, 132(45):16185– 16190, 11 2010. [182] Itamar Kimchi, Adam Nahum, and T. Senthil. Valence bonds in random quantum magnets:

Theory and application to ybmggao4. Phys. Rev. X, 8:031028, Jul 2018. [183] Lei Ding, Pascal Manuel, Sebastian Bachus, Franziska Grußler, Philipp Gegenwart, John Singleton, Roger D. Johnson, Helen C. Walker, Devashibhai T. Adroja, Adrian D. Hillier, and Alexander A. Tsirlin. Gapless spin-liquid state in the structurally disorder-free triangu-

lar antiferromagnet naybo2. Phys. Rev. B, 100:144432, Oct 2019. [184] Ji rˇ´ı Chaloupka, George Jackeli, and Giniyat Khaliullin. Zigzag magnetic order in the irid- ium oxide na2iro3. Phys. Rev. Lett., 110:097204, Feb 2013. [185] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. Vijay Shankar, Y. F. Hu, K. S. Burch, Hae- Young Kee, and Young-June Kim. α rucl : A spin-orbit assisted mott insulator on a − 3 honeycomb lattice. Phys. Rev. B, 90:041112, Jul 2014. [186] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Young-June Kim, H. Gretarsson, Yogesh Singh, P. Gegenwart, and J. P. Hill. Long-range magnetic ordering in na2iro3. Phys. Rev. B, 83:220403, Jun 2011. [187] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell, P. G. Radaelli, Yogesh Singh, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor. Spin waves and revised crystal structure of honeycomb iridate na2iro3. Phys. Rev. Lett., 108:127204, Mar 2012. [188] Yumi Kubota, Hidekazu Tanaka, Toshio Ono, Yasuo Narumi, and Koichi Kindo. Successive magnetic phase transitions in α rucl : Xy-like frustrated magnet on the honeycomb lattice. − 3 Phys. Rev. B, 91:094422, Mar 2015. [189] Sae Hwan Chun, Jong-Woo Kim, Jungho Kim, H. Zheng, Constantinos C. Stoumpos, C. D. Malliakas, J. F. Mitchell, Kavita Mehlawat, Yogesh Singh, Y. Choi, T. Gog, A. Al-Zein, M. Moretti Sala, M. Krisch, J. Chaloupka, G. Jackeli, G. Khaliullin, and B. J. Kim. Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate na2iro3. Nature Phys., 11:462 EP –, 05 2015. [190] M. Majumder, M. Schmidt, H. Rosner, A. A. Tsirlin, H. Yasuoka, and M. Baenitz. Anisotropic ru3+4d5 magnetism in the α rucl honeycomb system: Susceptibility, speciﬁc − 3 heat, and zero-ﬁeld nmr. Phys. Rev. B, 91:180401, May 2015. [191] Luke J. Sandilands, Yao Tian, Kemp W. Plumb, Young-June Kim, and Kenneth S. Burch. Scattering continuum and possible fractionalized excitations in α rucl . Phys. Rev. Lett., − 3 114:147201, Apr 2015. BIBLIOGRAPHY 177

[192] Stephen M. Winter, Kira Riedl, Pavel A. Maksimov, Alexander L. Chernyshev, Andreas Honecker, and Roser Valent´ı. Breakdown of magnons in a strongly spin-orbital coupled magnet. Nature Commun., 8(1):1152, 2017. [193] E. Onac, F. Balestro, L. H. Willems van Beveren, U. Hartmann, Y. V. Nazarov, and L. P. Kouwenhoven. Using a quantum dot as a high-frequency shot noise detector. Phys. Rev. Lett., 96:176601, May 2006. [194] Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deranlot, H. X. Yang, M. Chshiev, T. Valet, A. Fert, and Y. Otani. Giant spin hall effect induced by skew scattering from bismuth impurities inside thin film cubi alloys. Phys. Rev. Lett., 109:156602, Oct 2012. [195] Arne Brataas, Bart van Wees, Olivier Klein, Gregoire´ de Loubens, and Michel Viret. Spin insulatronics. Phys. Rep., 885:1–27, 2020. [196] Saul¨ Velez,´ Vitaly N. Golovach, Amilcar Bedoya-Pinto, Miren Isasa, Edurne Sagasta, Mikel Abadia, Celia Rogero, Luis E. Hueso, F. Sebastian Bergeret, and Felix` Casanova. Hanle magnetoresistance in thin metal films with strong spin-orbit coupling. Phys. Rev. Lett., 116:016603, Jan 2016. [197] Y. Shiomi, T. Ohtani, S. Iguchi, T. Sasaki, Z. Qiu, H. Nakayama, K. Uchida, and E. Saitoh. Interface-dependent magnetotransport properties for thin pt films on ferrimag- netic y3fe5o12. Applied Physics Letters, 104(24):242406, 2014. [198] B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee. Magnetoresistance and hall effect in a disordered two-dimensional electron gas. Phys. Rev. B, 22:5142–5153, Dec 1980. [199] B. L. Altshuler, A. G. Aronov, and P. A. Lee. Interaction effects in disordered fermi systems in two dimensions. Phys. Rev. Lett., 44:1288–1291, May 1980. [200] I. Paul, C. Pepin,´ B. N. Narozhny, and D. L. Maslov. Quantum correction to conductivity close to a ferromagnetic quantum critical point in two dimensions. Phys. Rev. Lett., 95:017206, Jul 2005. [201] Xuhui Wang, Jiang Xiao, Aurelien Manchon, and Sadamichi Maekawa. Spin-hall conductivity and electric polarization in metallic thin films. Phys. Rev. B, 87:081407, Feb 2013. [202] Henrik Bruus and Karsten Flensberg. Many-Body Quantum Theory in Condensed Matter Physics. Oxford University Press, Oxford, 2nd edition, 2002. [203] P. Schwab and R. Raimondi. Magnetoconductance of a two-dimensional metal in the presence of spin-orbit coupling. The European Physical Journal B - Condensed Matter and Complex Systems, 25(4):483–495, 2002. [204] B. L. Altshuler and A. G. Aronov. Contribution to the theory of disordered metals in strongly doped semiconductors. Journal of Experimental and Theoretical Physics, 77(5):2028, November 1979. BIBLIOGRAPHY 178

[205] Patrick A. Lee and T. V. Ramakrishnan. Disordered electronic systems. Rev. Mod. Phys., 57(2):287–337, 1985.

[206] S. V. Syzranov and J. Schmalian. Conductivity close to antiferromagnetic criticality. Phys. Rev. Lett., 109:156403, Oct 2012.

[207] Weinan Lin, Lei Li, Fatih Dogan,˘ Changjian Li, Hel´ ene` Rotella, Xiaojiang Yu, Bangmin Zhang, Yangyang Li, Wen Siang Lew, Shijie Wang, Wilfrid Prellier, Stephen J. Pennycook, Jingsheng Chen, Zhicheng Zhong, Aurelien Manchon, and Tom Wu. Interface-based tun- ing of rashba spin-orbit interaction in asymmetric oxide heterostructures with 3d electrons. Nature Communications, 10(1):3052, 2019.