New Forms of Ergodicity Breaking in Quantum Many-Body Systems Sanjay Moudgalya 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: B. Andrei Bernevig September 2020 © Copyright by Sanjay Moudgalya, 2020. All rights reserved. Abstract Generic isolated interacting quantum systems are believed to be ergodic, i.e. any simple initial state evolves to a thermal state at late-times, forming a barrier to the protection of quantum information. A sufficient condition for the thermalization of initial states in an isolated quantum system is the Eigenstate Thermalization Hypothesis (ETH). A complete breakdown of ETH is well-known in two kinds of systems: Integrable and Many-Body Localized, where none of the eigenstates satisfy it. In this dissertation, we introduce two new mechanisms of ergodicity breaking: Quantum Scars, and Hilbert Space Fragmentation. In both these cases, ETH-violating eigenstates coexist with ETH-satisfying ones, and thus the fate of an initial state under time-evolution depends on the properties of eigenstates it has weights on. We obtain the first analytical examples of quantum scars by solving for several excited states in a family of non-integrable quantum systems in one dimension: the AKLT models. These exact eigenstates include an infinite tower of states from the ground state to the highest excited state. The states in the middle of the spectrum obey a logarithmic scaling of entanglement entropy with system size, contrary to the volume-law scaling predicted by ETH. We further show the close connections between such quantum scarred models and Parent Hamiltonians of Matrix Product States, as well as connections to the phenomenon of eta-pairing known in the context of superconductivity in Hubbard models. Hilbert space fragmentation occurs in constrained systems with a center-of-mass or dipole moment conservation law, which naturally arises within a Landau level in quantum Hall systems or in systems subjected to a large electric field limit. We show that the Hilbert space of such systems fractures into several dynamically disconnected Krylov subspaces that are not distinguished by simple symmetries, and thus constitute a violation of conventional ETH. We show that ETH can be modified to apply to each connected subspace separately, and that large integrable and non-integrable subspaces can co-exist within the same system. iii Contents Abstract iii 0 Prelimanaries 1 0.1 Quantum Thermalization ....................... 6 0.2 Matrix Product States and Entanglement . 12 0.3 Outline of this thesis ......................... 18 1 Excited States of the AKLT Model 21 1.1 Introduction .............................. 21 1.2 The Spin-1 AKLT Model ....................... 23 1.3 Exact States .............................. 27 1.4 Exact Low energy Excited states of the Spin-1 AKLT Model . 29 1.5 Mid-Spectrum Exact States ..................... 35 1.6 1D Spin-S AKLT Models with S > 1 . 43 1.7 Conclusion ............................... 52 2 Entanglement of Quasiparticle Excited States 56 2.1 Introduction .............................. 56 2.2 MPS and MPO in the AKLT models . 59 2.3 MPO × MPS ............................. 66 2.4 Single-mode Excitations ....................... 74 2.5 Beyond Single-Mode Excitations ................... 81 2.6 Tower of States ............................ 87 2.7 Implications for ETH ......................... 92 2.8 Entanglement Spectra Degenarcies and Finite-Size Effects . 94 2.9 Conclusion ............................... 104 3 Quantum Scars from Matrix Product States 107 3.1 Introduction .............................. 107 3.2 MPS description of Quasiparticles . 109 3.3 Parent Hamiltonian . 113 3.4 Quantum Scarred Hamiltonians . 117 iv 3.5 New Families of AKLT-like Quantum Scarred Hamiltonians . 125 3.6 New Type of Quantum Scars: Two-Site Quasiparticle Operators . 129 3.7 Conclusions .............................. 139 4 Quantum Scars from Spectrum Generating Algebras 141 4.1 Introduction .............................. 141 4.2 Review of η-pairing in the Hubbard Model . 144 4.3 η-pairing on arbitrary graphs . 146 4.4 Examples of η-pairing . 149 4.5 Quantum Many-Body Scars from the Hubbard Model . 152 4.6 Connections to Quantum Scars . 157 4.7 D dimensions ............................. 159 4.8 RSGA and Quantum Scarred Models . 162 4.9 Conclusions .............................. 166 5 Hilbert Space Fragmentation 168 5.1 Introduction .............................. 168 5.2 Model and its symmetries . 170 5.3 Hamiltonian at 1/2 filling . 174 5.4 Krylov Fracture ............................ 178 5.5 Integrable subspaces . 184 5.6 Non-integrable subspaces and Krylov-Restricted ETH . 196 5.7 Quasilocalization from Thermalization . 200 5.8 Conclusions and Open Questions . 205 6 Approximate Quantum Scars in a Fractured Model 208 6.1 Introduction .............................. 208 6.2 Effective Spin-Chains and Constrained Hilbert spaces . 210 6.3 Symmetries and Non-integrability of the Effective Hamiltonians . 224 6.4 Quantum Many-Body Scars . 226 6.5 Forward Scattering Approximation . 228 6.6 Conclusions .............................. 236 Appendix A Algebra of dimers 238 A.1 Commutation relations . 238 A.2 Dimer basis states and scattering rules for the spin-1 AKLT model 240 A.3 Dimer basis scattering rules for spin-S AKLT basis states . 243 Appendix B Review of Matrix Product Operators 248 B.1 Matrix Product Operators . 248 B.2 Jordan normal form of block upper triangular matrices . 253 v Appendix C Embedding Quasiparticles using MPS Subspaces 259 C.1 Total Angular Momentum Eigenstates . 260 C.2 Examples of A and B subspaces for the AKLT-like MPS . 260 C.3 Single-Site Quasiparticle Exact Eigenstates in the MPS Language 262 C.4 SU(2) Multiplet of the Spin-2 Magnon for the AKLT chain . 265 C.5 Examples of A and Be subspaces for the Potts-like MPS . 267 Appendix D Eta pairing and RSGAs 269 D.1 Useful Identities ............................ 269 D.2 η-pairing with disorder and spin-orbit coupling . 270 D.3 Tower of States from (Restricted) Spectrum Generating Algebras 271 Appendix E Physical Origins of the Pair-Hopping Hamiltonian 275 E.1 Bloch Many-Body Localization . 276 E.2 Quantum Hall Effect . 287 References 315 vi Listing of figures 1.1 Level Statistics of the spin-1 AKLT Model . 25 1.2 AKLT Ground state. The big circles are physical spin-1 s and the smaller circles within the spin-1 s are spin-1/2 Schwinger bosons. Symmetric combinations of the Schwinger bosons on each site form the physical spin-1. The lines joining the Schwinger bosons repre- sent singlets. jGi with periodic boundary conditions . 25 1.3 Arovas A State Dimer Configuration . 32 1.4 Arovas B State Dimer Configurations . 33 1.5 Spin-2 Magnon Dimer Configuration . 35 1.6 Tower of States Dimer Configuration . 37 1.7 Destructive Interference of Spin-2 Magnons . 39 1.8 Position of the AKLT tower of states in the spectrum . 41 1.9 Spin-2 AKLT Ground State Dimer Configuration . 45 1.10 Spin-2 AKLT Arovas B State Dimer Configurations . 47 1.11 Spin-4 Magnon of Spin-2 AKLT Model . 51 1.12 Scattering Rules of the Spin-1 AKLT Model . 55 2.1 Spin-1 AKLT Ground State ..................... 59 2.2 Spin-2 AKLT Ground State ..................... 61 2.3 Entanglement entropy of AKLT eigenstates in the quantum number sectors with scar states ........................ 92 2.4 Entanglement Spectra of AKLT Towers of States . 102 5.1 Evidence of Krylov-Restricted ETH . 197 5.2 Quasilocalization from Thermalization . 201 6.1 Level Statistics of the ν = 2=5 Krylov subspace of the pair-hopping Hamiltonian .............................. 223 6.2 Hilbert Space Graph for the ν = 2=5 Krylov subspace of pair- hopping Hamiltonian . 224 6.3 Anomalous Dynamics of the Pair-Hopping Hamiltonian with ν = 2=5229 6.4 Approximate Scars of the Pair-Hopping Hamiltonian with ν = 2=5 231 vii 6.5 Stability of the ν = 2=5 scars . 233 A.1 Two types of singlet configurations around a bond fi; jg. 241 A.2 Types of non-singlet configurations around a bond fi; jg . 242 A.3 Scattering of S = 2 dimer configurations . 244 E.1 Bloch Many-Body Localization . 285 viii To my family, ix Acknowledgments This thesis would have not been possible without the enormous support of a lot of people, and I am woefully short of words to express my gratitude. Firstly, I would like to thank my advisor Andrei Bernevig for his guidance, encouragement, and support. Andrei’s vision and strong intuition shaped this thesis. His infectious enthusiasm and optimism taught me that physics could be a lot of fun, and his high standards of rigor constantly motivated me and greatly enhanced the clarity of my thoughts and scientific writing. I am also indebted to him for generously giving me the freedom and funding to pursue my own ideas and collaborations, due to which my grad school experience turned out to be exactly as I wanted it to be. Next, I would like to thank Nicolas Regnault, who played a role akin to my primary advisor’s. Nicolas’ humor and encouragement has made working on hard problems exciting. His patient comments have greatly improved my writing, pre- sentation, and computational skills. I am also extremely grateful for his invitation to spend a summer in ENS Paris, which has been one of the most fun and pro- ductive times during graduate school. I am also deeply indebted to Shivaji Sondhi for his guidance and for the many collaborations. Working closely with him on a wide variety of problems has been an incredible learning experience and helped me broaden my view of physics. In addition, his constant encouragement and close supervision right from the beginning has made navigating my way through grad school easier. I have further benefited a lot from several others in the physics community. I would particularly like to express my immense gratitude to Frank Pollmann for introducing me to research in condensed matter physics, and for his extraordinary support during graduate school applications without which this thesis would not exist.