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

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Transport, Criticality, and Chaos in Fermionic Quantum Matter at Nonzero Density Transport, criticality, and chaos in fermionic quantum matter at nonzero density a dissertation presented by Aavishkar Apoorva Patel to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2019 ©2019 – Aavishkar Apoorva Patel all rights reserved. Dissertation advisor: Professor Subir Sachdev Aavishkar Apoorva Patel Transport, criticality, and chaos in fermionic quantum matter at nonzero density Abstract This dissertation is a study of various aspects of metals with strong interactions between electrons, with a particular emphasis on the problem of charge transport through them. We consider the physics of clean or weakly-disordered metals near some quantum critical points, and highlight novel transport regimes that could be relevant to experiments. We then develop a variety of exactly-solvable lattice models of strongly interacting non-Fermi liquid metals using novel non-perturbative techniques based on the Sachdev-Ye-Kitaev models, and relate their physics to that of the ubiquitous “strange metal” normal state of most correlated- electron superconductors, providing controlled theoretical insight into the possible mechanisms behind it. Finally, we use ideas from the field of quantum chaos to study mathematical quantities that can provide evidence for the existence of quasiparticles (or the lack thereof) in quantum many-body systems, in the context of metals with correlated electrons. iii Contents 0 Introduction 1 0.1 The quantum mechanics of metals ............................... 1 0.2 “Strange” metals ........................................ 4 0.3 Metals beyond Fermi liquids .................................. 6 0.4 Many-body quantum chaos .................................. 14 1 Hyperscaling at the spin density wave quantum critical point in two dimensional metals 18 1.1 Introduction .......................................... 18 1.2 Model ............................................. 24 1.3 Optical conductivity σ(!) ................................... 28 1.4 T > 0 free energy ....................................... 37 1.5 Quantum Boltzmann Equation ................................. 40 1.6 Discussion ........................................... 51 2 DC resistivity at the onset of spin density wave order in two-dimensional metals 53 2.1 Introduction .......................................... 53 2.2 Symmetries and Noether currents ............................... 60 2.3 Memory Matrix Approach ................................... 61 2.4 Contributions to the DC Resistivity .............................. 63 2.5 Discussion ........................................... 68 3 Hydrodynamic flows of non-Fermi liquids: magnetotransport and bilayer drag 70 3.1 Introduction .......................................... 70 3.2 Hydrodynamic formalism ................................... 72 3.3 Magnetotransport in a single layer ............................... 73 3.4 Drag transport in bilayers ................................... 76 3.5 Discussion ........................................... 80 4 Shear viscosity at the Ising-nematic quantum critical point in two dimensional metals 82 4.1 Introduction .......................................... 82 4.2 Scaling arguments ....................................... 85 4.3 Field theory .......................................... 87 4.4 Optical shear viscosity ..................................... 90 4.5 Boltzmann equation and DC viscosity ............................. 94 4.6 Discussion ...........................................102 5 Magnetotransport in a model of a disordered strange metal 103 5.1 Introduction ..........................................103 5.2 Microscopic model .......................................108 5.3 Fate of the conduction electrons ................................110 5.4 Transport in a single domain ..................................118 5.5 Macroscopic transport via Effective-medium/Random-resistor theory . 129 iv 5.6 Discussion ...........................................136 6 Coherent superconductivity with a large gap ratio from incoherent metals 140 6.1 Introduction ..........................................140 6.2 Model 1 ............................................142 6.3 Model 2 ............................................145 6.4 Discussion ...........................................149 7 A critical strange metal from fluctuating gauge fields in a solvable random model 151 7.1 Introduction ..........................................151 7.2 Model and large-N limit ....................................154 7.3 Single-particle properties ...................................160 7.4 Thermodynamics ........................................169 7.5 Transport ............................................174 7.6 Discussion ...........................................177 8 Quantum chaos on a critical Fermi surface 179 8.1 Introduction ..........................................179 8.2 Model .............................................182 8.3 Scrambling and the Lyapunov exponent ............................185 8.4 The butterfly effect and energy diffusion . 191 8.5 Discussion ...........................................195 9 Quantum butterfly effect in weakly interacting diffusive metals 197 9.1 Introduction ..........................................197 9.2 Preliminaries ..........................................202 9.3 Many-body quantum chaos ..................................205 9.4 Additional considerations ...................................214 9.5 Two spatial dimensions ....................................219 9.6 Discussion ...........................................221 Appendix A Appendices to Chapter 1 223 A.1 Computation of hJyJyi ....................................223 A.2 Free Energy Computations ...................................233 A.3 Finite v and c ..........................................238 A.4 Boltzmann Equation Computations ..............................240 Appendix B Appendices to Chapter 2 251 B.1 Susceptibilities .........................................251 B.2 Computation of R(T ) .....................................253 B.3 Random Mass Computations .................................257 B.4 Vertex Correction for Inter Hot-Spot Scattering . 259 Appendix C Appendices to Chapter 3 263 C.1 Solution to linearized disordered magneto-hydrodynamic equations . 263 C.2 Magneto-thermal transport in the clean system . 267 C.3 Solution to hydrodynamic equations of the bilayer system . 268 C.4 Remarks on Coulomb drag in the ν = 1=2 quantum Hall state . 274 v Appendix D Appendices to Chapter 4 278 D.1 Optical viscosity: two-loop computations . 278 D.2 Relating conductivities and viscosities using Ward identities . 280 D.3 Contributions from the full Fermi surface . 285 Appendix E Appendices to Chapter 5 290 E.1 Effects of ‘Pair-hopping’ and bilinear terms on the marginal-Fermi liquid . 290 E.2 Boltzmann equation for the marginal-Fermi liquid . 292 Appendix F Appendices to Chapter 6 297 F.1 Derivation of gap equations ..................................297 F.2 Superconducting transition energetics .............................299 F.3 Real-time Dyson equations ..................................300 F.4 Gap equations for Model 2 ...................................303 Appendix G Appendices to Chapter 7 304 G.1 Higgs transition from the U(1) ACL to a Z2 ACL . 304 Appendix H Appendices to Chapter 8 309 H.1 Self energies ..........................................309 H.2 Wightman functions ......................................311 H.3 Higher order corrections ....................................312 H.4 Numerical methods ......................................313 H.5 Specific heat and thermal conductivity ............................315 Appendix I Appendices to Chapter 9 318 I.1 Outline of Feynman rules for the complex-time contour . 318 I.2 Absence of chaos in the non-interacting disordered metal . 320 References 345 vi Citations to previously published work Most of this dissertation has appeared in print elsewhere. Details are given below: • Chapter 1 has appeared as A. A. Patel, P. Strack and S. Sachdev, Physical Review B 92 (16), 165105 (2015). • Chapter 2 has appeared as A. A. Patel and S. Sachdev, Physical Review B 90 (16), 165146 (2014). • Chapter 3 has appeared as A. A. Patel, R. A. Davison and A. Levchenko, Physical Review B 96 (20), 205417 (2017). • Chapter 4 has appeared as A. Eberlein*, A. A. Patel* and S. Sachdev, Physical Review B 95 (7), 075127 (2017). • Chapter 5 has appeared as A. A. Patel, J. McGreevy, D.P. Arovas and S. Sachdev, Physical Review X 8 (2), 021049 (2018). • Chapter 6 has appeared as A. A. Patel and S. Sachdev, Physical Review B 98 (12), 125134 (2018). • Chapter 7 has appeared as A. A. Patel, M.J. Lawler and E.-A. Kim, Physical Review Letters 121 (18), 187001 (2018). • Chapter 8 has appeared as A. A. Patel and S. Sachdev, Proceedings of the National Academy of Sciences 114 (8), 1844-1849 (2017). • Chapter 9 has appeared as A. A Patel, D. Chowdhury, S. Sachdev and B. Swingle, Physical Review X 7 (3), 031047 (2017). *Equal contributions vii Some additional work performed by the author over the course of his graduate studies, but not included in this dissertation has appeared in • A. A. Patel, D. Chowdhury, A. Allais and S. Sachdev, Confinement transition to density wave order in metallic doped spin liquids, Physical Review B 93 (16), 165139 (2016). • A. A. Patel and A. Eberlein, Light induced enhancement of superconductivity via melting of competing
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