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The Sachdev-Ye-Kitaev Model and Matter Without Quasiparticles

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The Sachdev-Ye-Kitaev Model and Matter Without Quasiparticles The Sachdev-Ye-Kitaev model and matter without quasiparticles a dissertation presented by Wenbo Fu 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 2018 © 2018 - Wenbo Fu All rights reserved. Thesis advisor: Subir Sachdev Wenbo Fu The Sachdev-Ye-Kitaev model and matter without quasiparticles Abstract This thesis is devoted to the study of Sachdev-Ye-Kitaev (SYK) models, which describe matter without quasiparticles. The SYK model is an exactly solvable model that is hoped to bring insights into the understanding of strongly correlated materials. The main part discusses different aspects of SYK models. Chapter 2 presents numerical studies of the SYK models, shows its non-Fermi liquid nature and verifies the analytical predictions on two-point function and thermal entropies. Chapter 3 considers the thermody- namic and transport properties of the complex SYK models in higher dimensional general- izations, we show some universal relations obtained from SYK models which are consistent with holographic theories. Chapter 4 discusses a supersymmetric generalization of the SYK model. We show a different a scaling from the normal SYK model both analytically and numerically. We also discuss a different origin of the extensive zero temperature entropy in N = 2 theory. Chapter 5 considers a continuous set of boson-fermion SYK models with a tunable low-energy scaling dimension. From the explicit computation of out of time or- dered correlators, we find the theory is less chaotic in the free fermion limit. Chapter6 describes the phases of a solvable t-J model of electrons with an SYK-like form. We find a Z2 fractionalization picture where the deconfined phase is similar to the pseudo gap regime of underdoped cuprates, while the ‘quasi-Higgs’ phase is similar to the metallic phase of overdoped cuprates. iii Contents 1 Introduction 1 1.1 Preliminaries and motivations . 1 1.2 SYK 101 . 6 1.3 More about SYK . 12 1.4 Organization of the thesis . 14 2 Numerical studies on the SYK models 17 2.1 Introduction . 17 2.2 Model . 18 2.2.1 Large N limit for fermions . 18 2.2.2 Free energy and thermal entropy . 22 2.2.3 High temperature expansion . 24 2.3 Results . 27 2.3.1 Exact diagonalization for fermions . 27 2.3.2 Green’s function . 29 2.3.3 Entropy . 30 2.3.4 Entanglement entropy . 32 2.3.5 Out-of-time-order correlations and scrambling . 34 2.4 Discussion . 36 3 Thermoelectric transport in SYK models 38 3.1 Introduction . 38 3.2 Model . 40 3.2.1 Complex SYK model . 40 3.2.2 Large N saddle point . 41 3.3 Results . 43 3.3.1 Thermodynamics . 43 3.3.2 Effective action . 50 3.3.3 Transport . 55 iv 3.4 Discussion . 63 4 Supersymmetric SYK models 65 4.1 Introduction . 65 4.2 Model . 68 4.2.1 Definition of the model and the large N effective action . 68 4.2.2 Supersymmetry constraints . 73 4.2.3 Simple generalization . 73 4.3 Results . 77 4.3.1 Numerics from exact diagonalization . 77 4.3.2 N = 2 supersymmetry . 80 4.3.3 Four point function and the spectrum of operators . 83 4.4 Discussion . 87 5 A continuous set of boson-fermion SYK models 90 5.1 Introduction . 90 5.2 Model . 91 5.2.1 Definition of the model and the large N=M limit . 91 5.3 Results . 95 5.3.1 Imaginary time kernel . 95 5.3.2 Retarded kernel and chaos . 98 5.3.3 RG flow and the Schwarzian action . 100 5.3.4 Four-point functions . 105 5.3.5 Numerical imaginary time Green’s function correction . 112 5.3.6 Dephasing time . 114 5.4 Discussion . 115 6 Z2 fractionalized phases of a generalized SYK model 117 6.1 Introduction . 117 6.2 Model . 119 6.2.1 The SYK model from a t-J model . 119 6.2.2 The large N limit . 121 6.2.3 Gapless solutions . 124 6.2.4 Non-zero temperatures . 127 6.2.5 Gapped boson solution . 128 6.3 Results . 130 6.3.1 Composite operators . 130 v 6.3.2 Numerical Results . 133 6.3.3 Small gap scaling . 137 6.4 Discussion . 140 Appendices 142 A Appendix for chapter 2 143 A.1 SYK model for bosons . 143 B Appendix for chapter 3 146 B.1 Saddle point solution of the SYK model . 146 B.2 Kernel and spectrum . 148 B.3 Large q expansion of the complex SYK model . 154 B.4 Numerical solution of the SYK model . 159 B.5 Normal mode analysis of the SYK model . 162 B.6 Couplings in effective action of the SYK model . 165 B.7 Diffusion constants of the higher-dimensional SYK model . 168 B.8 Thermodynamics and transport . 169 C Appendix for chapter 4 174 C.1 The large qb limit of the supersymmetric SYK model . 174 D Appendix for chapter 5 177 D.1 The large q limit of the MN model . 177 E Appendix for chapter 6 179 E.1 Spectral functions . 179 E.2 High temperature expansion . 181 References 205 vi Citations to Previously Published Work Most of this thesis has appeared in print elsewhere. Details for particular chapters are given below. Chapter 2: • Wenbo Fu and Subir Sachdev “Numerical Study of fermion and boson models with infinite-range random interactions”, Physical Review B 94, 035135 (2016), arXiv:1603.05246 Chapter 3: • Partly based on: Richard A Davison, Wenbo Fu, Antoine Georges, Yingfei Gu, Kristan Jensen, Subir Sachdev, “Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and hologra- phy”, Physical Review B 95, 155131 (2017), arXiv:1612.00849 • Wenbo Fu, Yingfei Gu, Subir Sachdev and Grigory Tarnopolsky, “A study on complex SYK models”, in preparation Chapter 4: • Partly based on: Wenbo Fu, Davide Gaiotto, Juan Maldacena and Subir Sachdev “Supersymmetric Sachdev-Ye-Kitaev models”, Physical Review D 95, 026009 (2017), arXiv:1610.08917 Chapter 5: • Wenbo Fu, Chao-Ming Jian, Zhen Bi and Cenke Xu, “A continuous set of boson-fermion SYK models”, in preparation Chapter 6: • Partly based on: Wenbo Fu, Yingfei Gu, Subir Sachdev and Grigory Tarnopol- sky, “ Z2 fractionalized phases of a solvable, disordered, t − J model”, arXiv:1804.04130 Electronic preprints (shown in typewriter font) are available on the Internet at the following URL: http://arXiv.org vii To my parents. viii Acknowledgments There have been six years since I came to the beautiful Cambridge. It still seems I am quite new to this place. Although I know I have gained a lot during these years, but there are still much more unexplored. Cambridge is a magic place, because of the magical people live there. And I should say “Thanks” to many of them. Thank my advisor, Prof. Subir Sachdev. It is not only the knowledge that he has taught me but also the style that he works on physics that influences me a lot. Subir can always find the crucial aspects of physics problems and then try to solve them with the correct angle, not to mention that he always give the simple and deep physics picture for various of systems. Especially when I am the TA for Subir’s quantum phases of matter class, I am shocked by how he can explain things so clearly and unite different physics problems as a whole picture. Thank Subir for introducing me the interesting Sachdev-Ye-Kitaev model that this thesis is based on, I enjoyed a lot when studying and exploring. I would say there are many aspects that Subir has inspired me, his physics tastes and insights, his enthusiasm towards physics, his extraordinary technical skills and many more. I am pretty sure I will still benefit a lot from his influences in a wonderful way in the future. Thank my committee members: Prof. Bert Halperin, Prof. Philip Kim, and Prof. Ashvin Vishwanath. They can always give me quite useful feedbacks about my research work and many encouragements. Especially Bert’s course gives me a better understanding of the topological phases and Ashvin’s condensed matter journal club teaches me many different subjects. Thank many professors that have given wonderful courses that I have taken or sit at. I have taken three many-body physics courses including Subir’s, Prof. Patrick Lee’s and Prof. Xiao-Gang Wen’s. I can always gain different insights from the lectures and after-class discussions. I also learned a lot on CFTs and holographic dualities from classes given by Prof. Xi Yin and Prof. Hong Liu, which is quite helpful in the study of the Sachdev-Ye-Kitaev models. I also enjoyed a lot of many wonderful mathematical lectures or seminars from Prof. Hiro Lee Tanaka, Prof. Daniel Freed, Prof. Liang Kong and Prof. Jonathan Mboyo Esole. Although I still feel not quite natural in understanding pure math, I like it quite much. Thank Logan McCarty, Louis Deslauriers and David Morin for letting me teach in their courses and all the students I have communicated with, it also helps me better understand many standard physics questions. Thank Lisa Cacciabaudo, Carol Davis, Jacob Baran- des, Liz Alcock from the Harvard Physics Department, who have helped me on life outside research. Without their supports, life will definitely be much more complicated. Thank many friends in physics around here. These include the Subir group members: An- drea Allais, Shubhayu Chatterjee, Debanjan Chowdhury, Richard Davison, Andreas Eber- lein, Yingfei Gu, Yejin Huh, Janet Hung, Junhyun Lee, Andrew Lucas, Eun Gook Moon, ix Aavishkar Patel, Yang Qi, Mathias Scheurer, Julia Steinberg, Philipp Strack, Grisha Tarnopol- sky, Alex Thomson, Chong Wang, Seth Whitsitt, William Witczak-Krempa.
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