The actions of non-equilibrium systems and related matters Michael Joseph Landry Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2021 © 2021 Michael Joseph Landry All Rights Reserved Abstract The actions of non-equilibrium systems and related matters Michael Joseph Landry In this work, we develop an effective field theory program for many-body systems out of finite temperature equilibrium. Building on recent work, we combine powerful mathematical tools such as the Schwinger-Keldysh closed-time-path formalism, the coset construction, and Wilsonian effective field theory to construct novel actions that describe a wide range of many- body systems out of finite-temperature equilibrium. Unlike ordinary actions, these non- equilibrium actions account for dissipation and statistical and quantum fluctuations. The novel actions constructed include those for solids, supersolids, nematic liquid crystals, smectic liquid crystals in phases A, B, and C, chemically reacting fluids, quasicrystals, higher-form dual theories of superfluids and solids, and plasmas that can support large charge density. In order to construct these actions, we propose a new kind of coset construction with a total of four distinct types of inverse Higgs constraints. We extend the coset construction to account for higher-form symmetries and investigate the relationship between two kinds of ’t Hooft anomalies and spontaneous symmetry breaking. Table of Contents Acknowledgments .................................... v Dedication ........................................ vii Chapter 1: Introduction and Background ........................ 1 1.1 What is a non-equilibrium system? . 1 1.2 Why bother with effective field theory? . 3 1.3 Zoology and effective field theory . 5 1.4 Non-equilibrium effective actions . 8 1.4.1 KMS conditions and thermal equilibrium . 11 1.4.2 Rules for constructing non-equilibrium EFTs . 12 1.5 Diffusion action . 15 1.6 Relativistic hydrodynamics . 19 1.6.1 The standard approach . 19 1.6.2 The fluid worldvolume . 22 1.6.3 The action approach . 24 1.7 The meaning of retarded and advanced fields . 28 1.8 Organization of the thesis . 29 Chapter 2: The coset construction for non-equilibrium effective actions ........ 31 i 2.1 The zero-temperature coset construction: a review . 34 2.1.1 Inverse Higgs . 37 2.1.2 Zero-temperature superfluids: a simple example . 38 2.2 The non-equilibrium coset construction . 40 2.2.1 Goldstones at finite temperature . 41 2.2.2 The method of cosets . 42 2.3 Fluids and superfluids at finite temperature . 48 2.3.1 Fluids . 48 2.3.2 Superfluids . 53 2.4 Solids and supersolids at finite temperature . 55 2.4.1 Solids . 55 2.4.2 Supersolids . 58 2.5 Liquid crystals at finite temperature . 60 2.5.1 Nematic liquid crystals . 62 2.5.2 Smectic liquid crystals . 66 2.6 Conclusion . 71 Chapter 3: Higher-form symmetries and ’t Hooft anomalies in non-equilibrium systems 73 3.1 Introduction . 73 3.2 Higher-form symmetries: a review . 75 3.2.1 Superfluids at zero temperature . 78 3.2.2 Electromagnetism . 80 3.3 Non-equilibrium cosets and the classical limit . 81 ii 3.4 Non-equilibrium ’t Hooft anomalies . 85 3.4.1 Reactive fluids . 87 3.4.2 Second sound and its removal . 91 3.4.3 Dissipative solids and smectics without second sound . 94 3.4.4 Quasicrystals . 99 3.5 Higher-form symmetries and the coset construction . 105 3.5.1 The zero-temperature limit . 108 3.5.2 Gauge theories . 110 3.6 Higher-form symmetries and ’t Hooft anomalies . 116 3.6.1 Gauge fields and the Maurer-Cartan form . 116 3.6.2 Dual theories and mixed ’t Hooft anomalies . 119 3.6.3 Non-equilibrium higher-form ’t Hooft anomalies . 121 3.6.4 Quadratic examples . 123 3.7 Dual superfluids . 133 3.7.1 Zero temperature . 133 3.7.2 Finite temperature . 136 3.8 Dual solids . 138 3.8.1 Zero temperature . 138 3.8.2 Finite temperature . 141 3.9 Discussion . 144 Epilogue ......................................... 147 References ........................................ 150 iii Appendix A: Emergent gauge symmetries ....................... 157 Appendix B: Stückelberg tricks and the Maurer-Cartan form ............. 159 Appendix C: Explicit inverse Higgs computations ................... 162 C.1 Thermal inverse Higgs . 164 C.2 Unbroken inverse Higgs . 164 C.3 More on unbroken inverse Higgs . 166 Appendix D: Fluids and volume-preserving diffeomorphisms .............. 168 iv Acknowledgements There are may people who deserve acknowledgment and thanks both for my academic progress and my personal development. First I would like to thank my advisors, both official and unofficial. Thank you Alberto Nicolis and Lam Hui for your wonderful mentorship through out my graduate education. It has been a privilege learn from you and I have benefitted immensely from your uncommon talents. You managed to simultaneously teach me enormous amounts of physics while fos- tering an environment of independence. I truly could not have hoped for better advisors. Additionally, I thank Matteo Baggioli for the many projects and discussions we have shared. Your enthusiasm for physics is contagious and over the past year, you have broadened my un- derstanding of physics enormously. Austin Joyce deserves a big thank you from not only me but all of the other graduate students in the theory department. Your patience in explaining complicated concepts in a clear way is unparalleled. You have done so much for us and asked for so little in return. Additionally, I am extremely grateful to my undergraduate advisor, Cumrun Vafa for your enormous support, both professional and personal. Thank you to my high school physics teacher Michael Stark for your enthusiasm and encouragement during the earliest stages of my development as a physicist. In high school, I worked on a research project with Professor Dorian Goldfeld. I thank you for giving me my first introduction to research; you taught me an enormous amount and I am truly grateful for everything you have done for me. Lastly I would like to thank Professor Morgan May for encouraging my curiosity and indulging my many questions about the physical universe since I was a child. v A big thank you to my fiends Noah Bittermann, Roman Berrens, Guanhao Sun, Benjamin Gilbert, Steve Harrelson, and Farzan Vafa for the many discussions we have had. It has been a pleasure working with all of you. I have been extremely lucky to be surrounded by a loving and supportive family. None of this would have been possible without them. Thank you to both my American family for your unconditional love and support and thank you to my Russian family for accepting me as one of your own. A special thank you to my dad for teaching me calculus when I was twelve and to my mom for teaching me to read despite my best efforts. Thank you to Chris for being the best big brother I could have hoped for. And thank you to Anastasia for accompanying me, supporting me, and loving me every step of the way. Finally, thank you to Uncle Bob for everything (there is far too much to list here). Lastly, I am deeply grateful to the many doctors, nurses and other medical professionals without whom I would not be here. vi Dedication To my loving wife, Anastasia vii Chapter 1: Introduction and Background The aim of this thesis is to investigate the non-equilibrium dynamics of many-body systems from the perspective of effective field theory. In Chapter 1, we will review some important results derived in recent works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Chapters 2 and 3 contain original work that I completed as a graduate student. These chapters are primarily based on [29, 30] but also contain summaries of important results from [31, 32, 33]. Finally, as a graduate student, I also took part in [34, 35, 36], but in the interest of brevity, I will not discuss these works. 1.1 What is a non-equilibrium system? Perhaps a better question is: What is not a non-equilibrium system? There are two categories of physical systems that do not fall under the heading of non-equilibrium physics. The first includes equilibrium systems, most of which are in thermal equilibrium. The second includes zero-temperature systems, which are described by pure quantum states. In reality, both of these categories are idealizations; no system is ever in perfect equilibrium or exactly at zero temperature. In this sense, non-equilibrium physics includes all physical phenomena that are not subject to these idealized assumptions. The focus of this thesis will be the behavior of non-equilibrium phenomena with very many degrees of freedom. Such systems are often referred to as many-body systems. The behavior of many-body systems are often highly chaotic and complicated; they therefore resist simple analytic description. If, however, we consider only the behavior of these systems over long distance and time scales, the descriptions drastically simplify. The reason is that over sufficiently long scales, equilibrium is obtained in an approximate, local 1 sense. That is, much of the the complicated, chaotic behavior typically dies down over a time-scale known as the relaxation time. Thus, considering the behavior of systems on scales much longer than the relaxation time allows us to sweep much of the chaotic ugliness under the rug and define an effective description that involves only a small number of degrees of freedom. As a result, we may treat each (sufficiently large) local region as if it were in a state of approximate equilibrium.