Entanglement as a fundamental aspect of quantum mechanics plays a crucial role in many-body systems. Quantum dynamics, due to the intricate interplay between the locality of the interaction and the non-local nature of entanglement, greatly enriches the phenomena and brings the subject to the next complexity. In this dissertation, I will investigate entanglement and dynamics in low-dimensional systems from various perspectives. Chapter 1 examines the relationship between entanglement and topology in gapped quantum systems. We introduce a new framework called entanglement response, which offers a new perspective on the previously conjectured modular commutator formula that extracts the chiral central charge. Moreover, by incorporating global symmetries, we derive a new formula for the quantum Hall conductance. Chapter 2 extends our analysis of entanglement to mixed states. In particular, we inquire how entanglement can be used to define topological order in mixed states that are obtained by applying local decoherence to gapped ground states. This problem is naturally related to the active error correction, as we will address. Chapter 3 focuses on the dynamics of one-dimensional conformal field theories under specially engineered Floquet driving. We show that the Floquet dynamics support a heating phase that exhibits robust superuniversal features that rely only on conformal invariance. This heating phase is also shown to be useful in efficiently cooling. Chapter 4 delves into the entanglement dynamics of one-dimensional random unitary circuit doped with projective measurement. We focus on the limit where the measurement happens with a small probability. We will offer an explanation of the stability of the steady state by invoking the idea of quantum error correction.
Entanglement as a fundamental aspect of quantum mechanics plays a crucial role in many-body systems. Quantum dynamics, due to the intricate interplay between the locality of the interaction and the non-local nature of entanglement, greatly enriches the phenomena and brings the subject to the next complexity. In this dissertation, I will investigate entanglement and dynamics in low-dimensional systems from various perspectives. Chapter 1 examines the relationship between entanglement and topology in gapped quantum systems. We introduce a new framework called entanglement response, which offers a new perspective on the previously conjectured modular commutator formula that extracts the chiral central charge. Moreover, by incorporating global symmetries, we derive a new formula for the quantum Hall conductance. Chapter 2 extends our analysis of entanglement to mixed states. In particular, we inquire how entanglement can be used to define topological order in mixed states that are obtained by applying local decoherence to gapped ground states. This problem is naturally related to the active error correction, as we will address. Chapter 3 focuses on the dynamics of one-dimensional conformal field theories under specially engineered Floquet driving. We show that the Floquet dynamics support a heating phase that exhibits robust superuniversal features that rely only on conformal invariance. This heating phase is also shown to be useful in efficiently cooling. Chapter 4 delves into the entanglement dynamics of one-dimensional random unitary circuit doped with projective measurement. We focus on the limit where the measurement happens with a small probability. We will offer an explanation of the stability of the steady state by invoking the idea of quantum error correction.
As Schroedinger already recognised one century ago, entanglement is at the core of quantum mechanics. Nowadays it turns out to be the fundamental notion behind many quantum phenomena, from quantum algorithms to gravity, passing by critical phenomena and topological phases of matter, triggering unexpected connections between apparently far branches of physics. At the center of all these ideas, we find the (Rényi) entanglement entropies, which are powerful entanglement measures that provide fundamental insights into the investigated system or theory. This motivates the development of techniques for determining it, such as the replica trick: its implementation via the path-integral finds a wide place in this thesis. For example, we develop an efficient strategy to compute a generalised version of the Rényi entropies for all the eigenstates of a (1+1)-dimensional conformal field theory. This represents the starting point for a simulation scheme ideal to compute the entanglement in more generic (1+1)-dimensional quantum field theories (QFTs), e.g. after a quench in the sine-Gordon field theory. The study of entanglement also intertwines with another pillar of modern physics, i.e. symmetries and how their presence influences the properties of a system. Given the interest in this connection, this thesis addresses the question of how the entanglement splits into the different sectors of an internal symmetry. We approach the problem first in the QFT context, both for the free Dirac and complex scalar fields in two-dimensional spacetime, which have an abelian conserved charge, and systems having an internal Lie group symmetry to tackle the non-abelian case. Another typical framework in which we study the symmetry resolution of entanglement is lattice models, where different techniques can be exploited in order to derive exact results, ranging from the corner transfer matrix for gapped integrable systems to the connection between quadratic lattice Hamiltonians and their two-point correlation functions. The symmetry resolution also concerns other entanglement measures, namely, we analyse the behaviour of the operator entanglement, i.e. a key quantifier of the complexity of an operator, the symmetry-resolved mutual information, the effect of symmetries on entanglement negativity. The latter quantity is a genuine measure of quantum correlations in mixed states and a consistent part of the thesis is about this subject. For example, we study its time evolution after a quench and we provide an operatorial characterisation for entanglement in mixed states, which we dub negativity Hamiltonian.
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.
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.
In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff.
Understanding the fate of many-body quantum physics in the presence of noise is of both practical and fundamental interest. Practically, all quantum systems (especially near-term experiments) are interacting with their environment, and so understanding the role that noise plays is crucial to making predictions of observations. Fundamentally, one can find novel physical phenomena in open systems that exists only out of equilibrium. This motivates trying to get a basic understanding of mixed state phases of matter. This dissertation will be in two parts, linked by the common theme of understanding the dynamics and steady states of open, many-body quantum systems. We will first ask about dissipative state preparation, or thinking about how one can use openness in a quantum system to engineer interesting quantum states. We will present a number of exact solutions for spins, free bosons, and free fermions that have unique properties in the steady state, be they highly entangled or topological in nature. Then, we will discuss their dynamics, and present fundamental limits on the time it takes to prepare highly entangled states under local Markovian channels, and also ways that one can use additional ingredients to evade these bounds. The second part will focus on a novel technique to simulate the dynamics of open spin-1/2 systems whose coupling to the environment will formally make them interacting. This technique relies on encoding the many-body density matrix into a stochastic covariance matrix of the Jordan-Wigner fermions which we show can be sampled from efficiently. We use this technique to uncover rich phenomena outlined in a number of many-body systems, as well as provide certain situations in which the stochastic observables can be analytically re-averaged to get closed form solutions for the entire interacting system.