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Symmetry, Topology and Entanglement in Quantum Many-Body Systems

Liu, Shang

Published Jan 1, 2021• 0 citations
OpenAlex

Abstract

Symmetry, topology and entanglement are three indispensable concepts for our understanding of quantum many-body systems. In this dissertation, we demonstrate the use and interplay of these concepts with four distinct topics. The first part focuses on two-dimensional (2D) symmetry protected topological (SPT) phases with lattice rotation symmetries. We start by discussing a prototypical example that exhibits interesting topological responses, and then investigate general aspects of rotation protected topology. In the second part, we report on a Hartree-Fock calculation of possible symmetry-broken phases in magic angle bilayer graphene at charge neutrality, assuming that the valley U(1) symmetry is preserved. Our solutions include both correlated insulators and nematic topological semimetals, whose relative stability can be tuned by weak strains. In the third part, inspired by the recent studies on gapless SPT states, we examine the fate of magnetic impurities at 2+1D quantum critical points. We introduced a series of large-N solvable impurity models which, as a consequence of topology, demonstrate intermediate coupling fixed points. In the last part, we develop a diagrammatic method to compute the entanglement properties of random mixed states which are obtained by partial tracing random pure states and are relevant for a variety of problems ranging from chaotic quantum dynamics to black hole physics.

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As condensed matter theorists, we always try to seek new quantum phases of matter that are not possible in classical physics. In this dissertation, I discussed a new type of quantum disordered phases known as symmetry protected topological (SPT) phases, which is a generalization of the topological insulator to interacting fermion or boson/spin systems with various symmetries. In the first part of this thesis, a nonlinear σ-model (NLσM) field theory is introduced as a powerful tool to describe the properties of the bosonic SPT phases. Secondly, we want to answer the question of how to detect the SPT states from their bulk properties. Introducing gauge fields was shown to be an effective theoretical tool to study bulk properties of SPT phases. Furthermore, we investigated anyon and loop statistics of gauged SPT states in the framework of NLσM. We also designed a new numerical probe, so-called strange correlator, which can distinguish SPT states from trivial states based on the bulk wavefunction on a closed manifold. Thirdly, several aspects of surface states of SPT phases are discussed. 1. A surface phase transition of 3d topological insulator is studied through a new controlled expansion method with the help of the recently discovered fermion-vortex duality. 2. A new strongly interacting conformal field theory on the surface of 3d bosonic SPT state is also found by a controlled renormalization group calculation. 3. we made a connection between the surface of SPT phase and the Lieb-Schultz-Mattis (LSM) theorem, which enables us to identity the SU(N) and SO(N) spin systems that permit a featureless ground state in 2d and 3d. Finally, we proposed the first experimental realization of bosonic SPT state in dimension higher than 1. We established a general relation between interacting multi-layer fermionic SPTs and bosonic SPT with the same symmetry, which motivates a proposal of realizing 2+1d bosonic SPT phase in bilayer graphene system.

•Jan 1, 2017•0 citations
Symmetry, topology and entanglement are three indispensable concepts for our understanding of quantum many-body systems. In this dissertation, we demonstrate the use and interplay of these concepts with four distinct topics. The first part focuses on two-dimensional (2D) symmetry protected topological (SPT) phases with lattice rotation symmetries. We start by discussing a prototypical example that exhibits interesting topological responses, and then investigate general aspects of rotation protected topology. In the second part, we report on a Hartree-Fock calculation of possible symmetry-broken phases in magic angle bilayer graphene at charge neutrality, assuming that the valley U(1) symmetry is preserved. Our solutions include both correlated insulators and nematic topological semimetals, whose relative stability can be tuned by weak strains. In the third part, inspired by the recent studies on gapless SPT states, we examine the fate of magnetic impurities at 2+1D quantum critical points. We introduced a series of large-N solvable impurity models which, as a consequence of topology, demonstrate intermediate coupling fixed points. In the last part, we develop a diagrammatic method to compute the entanglement properties of random mixed states which are obtained by partial tracing random pure states and are relevant for a variety of problems ranging from chaotic quantum dynamics to black hole physics.
This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids. The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.
This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids. The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.
Quantum phenomena at macroscopic scales are fascinating, given that quantum mechanical effects usually manifest only at atomic scales due to their fragile nature. Enriched with symmetry, topology, and entanglement, quantum matters exhibit exotic emergent properties that defy our classical intuitions. This dissertation is devoted to the study of quantum matters, consisting of three distinct topics. The first part studies quantum phases of matter and their phase transitions, with an emphasis on the role of symmetry and their exotic natures. Employing both analytical and numerical methods, we study the deconfinement physics in quantum Hall systems and quantum magnets. The second part focuses on the physics of Moir\'e systems in two-dimensional materials that exhibit interesting single-particle as well as correlated many-body physics. Using a continuum model description, we illuminate several recent experimental findings in twisted double bilayer graphenes. The last part touches on a more unconventional topic, non-Hermitian systems. We provide mathematical foundations to classify their topological structure as well as the potential connection to anomalous or driven systems.
We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators (TBIs) to interacting many-particle systems known as symmetry-protected topological (SPT) phases. As with the TBIs, these states have a bulk gap and no exotic excitations but have nontrivial surface states that are protected by symmetry. We describe the various possible phases and their properties in three-dimensional systems with realistic symmetries. We develop many key ideas for the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.
We study several aspects of the realization of global symmetries in highly entangled phases of quantum matter. Examples include gapped topological ordered phases, gapless quantum spin liquids, and non-Fermi liquid phases. An insightful window into such phases is provided by recent developments in the theory of short-ranged entangled symmetry protected topological (SPT) phases. First, they generate useful no-go constraints on how global symmetry may be implemented in a highly entangled phase. Possible symmetry implementation in gapped topological phases and some proposed gapless spin/Bose liquids are examined in this light. We show that some previously proposed spin liquid states for 2D quantum magnets do not in fact have consistent symmetry implementation unless they occur as the surface of a 3D SPT phase. A second SPT-based insight into highly entangled states is the development of a view point of such states as SPT phases of one of the emergent excitations. We describe this in the specific context of time-reversal symmetric 3D U(1) quantum spin liquids with an emergent photon. Different such spin liquids are shown to be equivalent to different SPT insulating phases of the emergent monopole excitation of such phases. The highly entangled states also in turn enrich our understanding of SPT phases. We use the insights obtained from our results to provide an explicit construction of bosonic SPT phases in 3D in a system of coupled layers. This includes construction of a time-reversal symmetric SPT state that is not currently part of the cohomology classification of such states.
As condensed matter theorists, we always try to seek new quantum phases of matter that are not possible in classical physics. In this dissertation, I discussed a new type of quantum disordered phases known as symmetry protected topological (SPT) phases, which is a generalization of the topological insulator to interacting fermion or boson/spin systems with various symmetries. In the first part of this thesis, a nonlinear σ-model (NLσM) field theory is introduced as a powerful tool to describe the properties of the bosonic SPT phases. Secondly, we want to answer the question of how to detect the SPT states from their bulk properties. Introducing gauge fields was shown to be an effective theoretical tool to study bulk properties of SPT phases. Furthermore, we investigated anyon and loop statistics of gauged SPT states in the framework of NLσM. We also designed a new numerical probe, so-called strange correlator, which can distinguish SPT states from trivial states based on the bulk wavefunction on a closed manifold. Thirdly, several aspects of surface states of SPT phases are discussed. 1. A surface phase transition of 3d topological insulator is studied through a new controlled expansion method with the help of the recently discovered fermion-vortex duality. 2. A new strongly interacting conformal field theory on the surface of 3d bosonic SPT state is also found by a controlled renormalization group calculation. 3. we made a connection between the surface of SPT phase and the Lieb-Schultz-Mattis (LSM) theorem, which enables us to identity the SU(N) and SO(N) spin systems that permit a featureless ground state in 2d and 3d. Finally, we proposed the first experimental realization of bosonic SPT state in dimension higher than 1. We established a general relation between interacting multi-layer fermionic SPTs and bosonic SPT with the same symmetry, which motivates a proposal of realizing 2+1d bosonic SPT phase in bilayer graphene system.