Quantum mechanical systems with holographic duals Alexey Milekhin A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Professor Juan Maldacena September 2020 c Copyright by Alexey Milekhin, 2020. All rights reserved. Abstract This thesis is devoted to studying quantum mechanical systems with gravity duals. It is in- teresting to study holographic correspondence for quantum mechanical systems since we have much more theoretical control over them compared to quantum field theories. At the same time, gravity duals to quantum mechanical systems are quite rich as they can include black holes and wormholes. Chapter 2 is based on work [1] with J. Maldacena and studies aspects of gauge symmetry in Banks-Fishler-Shenker-Susskind(BFSS) model. In the original formulation it includes gauged SU(N) symmetry. However, we argued that non-singlet states are separated by a finite gap from the ground state. Therefore, gauging SU(N) symmetry is not important at low energies. Chapter 3 is based on paper [2] with A. Almheiri and B. Swingle. It is dedicated to study- ing thermalization dynamics of systems with gravity duals. We argued that average null en- ergy condition(ANEC) in the bulk leads to a universal bound on the total amount of energy exchange between two quantum systems. We study this bound perturbatively and in Sachdev- Ye-Kitaev(SYK) model at arbitrary coupling. As a byproduct, we studied the non-equilibrium dynamics of SYK, both analytically and numerically. Chapter 4 is based on paper [3] with J. Maldacena. We study wormhole formation in SYK model in real time. We start from a high temperature state, let it cool by coupling to a cold bath and numerically solve for the large N dynamics. Our main result is that the system forms a wormhole by going through a region with negative specific heat, taking time that is independent of N. Chapter 5 is based on paper [4] with I. Klebanov, F. Popov and G. Tarnopolsky. This paper is dedicated to studying various spectral properties of large N melonic tensor models. They have the same large N limit as SYK model, but unlike SYK they do not include disorder average. We find the exact expression for the number of singlet states and derive various bounds on energies. iii Acknowledgements Chapter 3 of this thesis was presented at the program Universality and ergodicity in quantum many-body systems in Stony Brook University. Chapter 4 was presented at Quantum mat- ter/Quantum field theory seminar in Harvard University. First of all, I am indebted to my adviser, Juan Maldacena, for his constant guidance and patience through many different research projects and topics and also for sharing his incredible physical intuition with me. I am especially grateful to Igor Klebanov for a long-term collabo- ration, his creativity and open-mindedness and his willingness to be in the thesis committee. I must also mention my undergraduate adviser, Alexander Gorsky, for introducing me to quantum field theory, a lot of scientific discussions and his continual support for almost ten years. Also I am grateful to my co-authors throughout my graduate years: Ahmed Almheiri, Kse- nia Bulycheva, Bruno Le Floch, Sergei Nechaev, Fedor Popov, Nikita Sopenko, Brian Swingle, Grigory Tarnopolsky and Wenli Zhao. All of them have their unique approach to physics and research in general, and I feel that each of these collaborations have made me better as a scientist. I am thankful to Simone Giombi for being the second reader of the thesis and Lyman Page for being in the thesis committee. I would like to thank all the people I have had discussions with for their ideas, encouragement, criticism, comments and time: Alexander Abanov, Semeon Artamonov, Alexander Avdoshkin, Ilya Belopolski, Damon Binder, Yiming Chen, Raffaele D'Agnolo, Nikolay Dedushenko, Xi Dong, Yale Fan, Hrant Gharibyan, Akash Goel, Luca Iliesiu, Christian Jepsen, Ziming Ji, Jiaqi Jiang, Alex Kamenev, Dmitri Kharzeev, Sergey Khilkov, Vladimir Kirilin, Zohar Komargodski, Ho Tat Lam, Henry Lin, Mariangela Lisanti, Donald Marolf, Victor Mikhaylov, Baur Mukhamet- zhanov, Nikita Nekrasov, Silviu Pufu, Boris Runov, Andrey Sadofyev, Stephen Shenker, Dou- glas Stanford, Nikolay Sukhov, Joaquin Turiaci, Jacobus Verbaarschot, Herman Verlinde, Zhenbin Yang, Przemek Witaszczyk, Wenli Zhao and Yunqin Zheng. I would like to give special thanks to my friends: Ksenia Bulycheva, Ivan Danilenko, Maria Danilenko, Antonio Alfieri and Fedor Popov for their constant support throughout my iv PhD years. I am sure without them I would not have been able to carry through. I am grateful to my girlfriend, Courtney King, for her support, patience, creativity and proofreading. Last, but not least, I want to thank my friends: Vasyl Alba, Lev Arzamasskiy, Levon Avanesyan, Tyler Boyd-Meredith, David Buniatyan, Erin Ellis, Himanshu Khanchandani, Misha Ivanov, Natasha Klimova, Roman Kolevatov, Artem Kotelskiy, Alexey Lavrov, Steven Li, Maksim Litske- vich, Elizaveta Mankovskaya, Vadim Munirov, Congling Qiu, Sergey Ryabichko, Elvira Say- futyarova, Charlie Stibitz, Alina Tokmakova, Eva Troje, Yury Ustinovskiy, Anibal Velozo and Andrei Zeleneev for making Princeton feel like home. v To my mother. vi Contents Abstract . iii Acknowledgements . iv 1 Introduction 1 1.1 An overview . .1 1.2 The D0 brane matrix model . .2 1.2.1 The matrix model . .3 1.2.2 The gravity dual . .4 1.3 A brief review of Sachdev{Ye{Kitaev model . .6 1.4 Klebanov{Tarnopolsky tensor model . 10 2 To gauge or not to gauge? 14 2.1 Introduction . 14 2.2 The ungauged model . 17 2.2.1 The size of the matrix versus the size of the Einstein gravity region . 18 2.2.2 Lack of supersymmetry . 18 2.2.3 Supersymmetric version of the ungauged model . 19 2.2.4 Relation to Wilson loop insertions . 20 2.3 Gravity duals of non-singlets . 21 2.3.1 Exploring the large X region . 23 2.3.2 Adjoint energies at weak coupling in the BMN matrix model . 25 2.3.3 Spectrum above the minimum . 26 vii 2.3.4 The free energy . 29 2.4 Deconfinement and the eigenvalues Polyakov loop holonomy . 31 2.5 Further comments . 35 2.5.1 Is there a bulk SU(N) gauge field associated to the SU(N) global symme- try of the ungauged model? . 35 2.5.2 Are there gauge fields on brane probes? . 36 2.5.3 The ungauged model and M-theory . 36 2.5.4 Physical realizations . 37 2.6 Conclusions . 37 3 SYK thermalization and universal constraints on energy flow 39 3.1 Introduction . 39 3.1.1 Summary of results . 41 3.2 Bounds on energy dynamics . 44 3.2.1 Perturbative bound . 44 3.2.2 Multi-operator couplings . 46 3.2.3 Relation to energy conditions in holography . 47 3.3 Thermalization in SYK . 51 3.3.1 Coupling to a bath . 51 3.3.2 Equilibrium . 54 3.3.3 Energy flux . 56 3.3.4 Very early time . 57 3.3.5 Early time . 57 3.3.6 Intermediate time . 60 3.3.7 Late time: approach to equilibrium and black hole evaporation . 67 3.3.8 Checking the bound numerically . 70 3.3.9 Comparison to exact finite N calculations . 71 3.4 Discussion . 73 viii 4 SYK wormhole formation in real time 75 4.1 Introduction and Summary . 75 4.1.1 Motivation . 75 4.1.2 Wormhole formation in SYK . 76 4.1.3 Equilibrium thermodynamics . 77 4.1.4 Gravity picture . 79 4.2 The two coupled SYK model and its thermodynamics . 81 4.2.1 Definition and properties of the ground state . 81 4.2.2 Perturbation theory at high temperature . 84 4.2.3 Low temperature thermodynamics using the Schwarzian . 87 4.3 Real time results . 91 4.3.1 Coupling to a bath . 91 4.3.2 Kadanoff–Baym equations . 93 4.3.3 Forming the wormhole . 96 4.3.4 Time to form the wormhole . 101 4.4 Two coupled black holes in gravity . 104 4.4.1 High temperature phase . 105 4.4.2 Low temperature phase . 107 4.4.3 Comparison with the SYK model . 108 4.5 Conclusion . 110 5 Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics 112 5.1 Introduction and Summary . 112 5.2 The rank-3 tensor model and its symmetries . 116 5.3 Energy bounds for the O(N1) × O(N2) × O(N3) model . 119 5.3.1 Basic bounds . 119 5.3.2 Refined bounds . 123 5.4 Sigma model and energy gaps . 127 ix 5.5 Counting singlet states . 129 5.5.1 Number of singlets for large N ........................ 134 5.5.2 Anomalies . 136 5.6 Solution of some fermionic matrix models . 137 5.6.1 The O(N1) × O(N2) model . 138 5.6.2 The SU(N1) × SU(N2) × U(1) model . 139 5.6.3 The O(N1) × O(N2) × U(1) model . 142 A Details of BFSS calculations 149 A.1 Details of the perturbative computations . 149 A.1.1 Non singlets in the BMN matrix model . 149 A.1.2 BFSS model . 153 A.1.3 Goldstone modes and SU(N) rotators for the BMN model vacua . 159 A.2 Analyzing the motion of a folded string . 162 A.3 Scaling properties of the solution and the action . 164 B SYK technicalities 167 B.1 Numerical setup for KB equations . 167 B.2 Energy flux from KB equations . 170 B.3 Locating the peak . ..