The dissertation includes two parts. In Part I, we study non-equilibrium phenomena in various models associated with global quantum quench. It is known that local quantities, when subjected to global quantum quench across or approaching critical points, exhibit a variety of universal scaling behaviors at various quench rates. To investigate if similar scaling holds for non-local quantities, we consider the scaling behavior of circuit complexity under quantum quench across the critical massless point in Majorana fermion field theory of the one-dimensional integrable transverse field Ising model and find it obeys such scaling. To investigate if similar scaling holds for non-relativistic theories, we test various solvable critical quantum quench protocols in a theory of fermions in a harmonic oscillator potential and find local quantities as well as entanglement entropy obeys different scaling behaviors at different quench rates. We study quantum quench in the c=1 matrix model which is holographically dual to two-dimensional string theory. Unlike higher dimensional holographic setups where quantum quench leads to black holes, the emergent spacetime in this model generically develops cosmological singularities at late times. In Part II, we expand the proposal that target space entanglement provides a precise notion of entanglement in the bulk gravitational duals of Dp brane theories, which was shown in a gauge fixed formalism. We developed a gauge invariant description of target space entanglement in these theories and derived path integral expressions for the entanglement entropy which can be used in numerical calculations.
The dissertation includes two parts. In Part I, we study non-equilibrium phenomena in various models associated with global quantum quench. It is known that local quantities, when subjected to global quantum quench across or approaching critical points, exhibit a variety of universal scaling behaviors at various quench rates. To investigate if similar scaling holds for non-local quantities, we consider the scaling behavior of circuit complexity under quantum quench across the critical massless point in Majorana fermion field theory of the one-dimensional integrable transverse field Ising model and find it obeys such scaling. To investigate if similar scaling holds for non-relativistic theories, we test various solvable critical quantum quench protocols in a theory of fermions in a harmonic oscillator potential and find local quantities as well as entanglement entropy obeys different scaling behaviors at different quench rates. We study quantum quench in the c=1 matrix model which is holographically dual to two-dimensional string theory. Unlike higher dimensional holographic setups where quantum quench leads to black holes, the emergent spacetime in this model generically develops cosmological singularities at late times. In Part II, we expand the proposal that target space entanglement provides a precise notion of entanglement in the bulk gravitational duals of Dp brane theories, which was shown in a gauge fixed formalism. We developed a gauge invariant description of target space entanglement in these theories and derived path integral expressions for the entanglement entropy which can be used in numerical calculations.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
It was found, by studying black holes, that the compatibility of the theory of gravity and the laws of quantum mechanics demands that the universe must act like a hologram. That is, all the information inside a region of the universe should be encoded on a so-called holographic screen of one-lesser dimension. This is termed the holographic principle. In a generic spacetime, such as an expanding universe, holographic screens depend on the choice of an observer, which is consistent with the notion that the observer is part of the system in cosmology. In the first part of this dissertation, we study the observer-dependence of holographic screens. This will help us understand how the fundamental description of the universe depends on the choice of the observer. Furthermore, we study the dynamics of the holographic screens and their geometry in the first part of this dissertation.An example where the holographic principle is manifest is the AdS-CFT correspondence. This correspondence connects quantum information quantities, like entanglement and complexity, to geometric quantities, like area and volume, respectively. In the second and the third part of this dissertation, we use the AdS-CFT correspondence as a tool to calculate the entanglement entropy and the computational complexity of a quenched CFT state. Studying the time evolution of the entanglement entropy following a quantum quench teaches us how a CFT state thermalizes. On the other hand, studying the time evolution of the computational complexity allows us to check the validity of various recent conjectures involving black holes and complexity.
In the first part of this dissertation, we study the dynamics of isolated and clean quantum systems out of equilibrium. We initially address the Kibble-Zurek (KZ) problem of determining the dynamical evolution of a system close to its critical point under slow changes of a control parameter. We formulate a scaling limit in which the nonequilibrium behavior is universal and discuss the universal content. We then report computations of some scaling functions in model Gaussian and large-N problems. Next, we apply KZ scaling to topologically ordered systems with no local order parameter. In the examples of the Ising gauge theory and the SU(2)k phases of the Levin-Wen models, we observe a slow, coarsening dynamics for the string-net that underlies the physics of the topological phase at late times for ramps across transitions that reduce topological order. We conclude by studying quenches in the quantum O(N) model in the infinite N limit in varying spatial dimensions. Despite the failure to equilibrate owing to an infinite number of emergent conservation laws, the qualitative features of late time states following quenches is predicted by the equilibrium phase diagram. In the second part of this dissertation, we explore the relationship between entanglement and topological order in fractional quantum Hall (FQH) phases. In 2008, Li and Haldane conjectured that the entanglement spectrum (ES), a presentation of the Schmidt values of a real space cut, reflects the energy spectrum of the FQH chiral edge. Specifically, both spectra should have the same quasi-degeneracy of eigenvalues everywhere in the phase. We offer an analytic, microscopic proof of this conjecture in the Read-Rezayi sequence of model states. We further identify a different ES that reflects the bulk quasihole spectrum and prove a bulk-edge correspondence in the ES. Finally, we show that the finite-size corrections of the ES of the Laughlin states reveal the fractionalization of the underlying quasiparticles.
Abstract We investigate a class of exactly solvable quantum quench protocols with a finite quench rate in systems of one dimensional non-relativistic fermions in external harmonic oscillator or inverted harmonic oscillator potentials, with time dependent masses and frequencies. These hamiltonians arise, respectively, in harmonic traps, and the c = 1 Matrix Model description of two dimensional string theory with time dependent string coupling. We show how the dynamics is determined by a single function of time which satisfies a generalized Ermakov-Pinney equation. The quench protocols we consider asymptote to constant masses and frequencies at early times, and cross or approach a gapless potential. In a right side up harmonic oscillator potential we determine the scaling behavior of the one point function and the entanglement entropy of a subregion by obtaining analytic approximations to the exact answers. The results are consistent with Kibble-Zurek scaling for slow quenches and with perturbation calculations for fast quenches. For cis-critical quench protocols the entanglement entropy oscillates at late times around its initial value. For end-critical protocols the entanglement entropy monotonically goes to zero inversely with time, reflecting the spread of fermions over the entire line. For the inverted harmonic oscillator potential, the dual collective field description is a scalar field in a time dependent metric and dilaton background.
In this PhD thesis we investigate some properties of one-dimensional quantum systems, focusing on two important aspects of integrable models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will therefore be devoted to the study of the entanglement entropy in one-dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting theory with non perturbative solutions. We derive its bipartite Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap is analysed, both analytically and numerically. In particular it is worth mentioning that the numerical analysis of the entropies of the XYZ model presented in this dissertation has never been published in literature. In the second part of the thesis we study the dynamics of correlators after a quantum quench, preparing the system in a squeezed coherent initial state. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, the latter studied by means of the semi-classical theory, the former by a form-factor approach. Moreover in the last chapter we outline a general result about the dynamics of correlation functions of local observables after a quantum quench. In particular we show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model at long times (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations). For the Transverse Field Ising Chain this result means that its dynamics is described by a Generalized Gibbs Ensemble.
The transverse field Ising model of N fully connected spins provides a mathematical accessible, yet non-trivial, system to investigate the non-thermal dynamics after a sudden quantum quench. The focus in this thesis is three-fold. First, in a 1/N expansion the dynamics of the mean magnetization within its variance is obtained analytically in the large N limit. The variance constitutes a leading order correction to the mean field limit at infinite N. While mean field theories are one of the most accessible approximations to deal with the quantum complexity, its validity in time out of equilibrium has to be checked on a case by case basis. By studying the dynamics of the variance, we see that the mean field approximation can break down at surprisingly early times scaling with the square root of N. The underlying mechanism in the fully connected Ising model is identified as a dephasing effect between nearby effective orbits. The dynamics of the magnetization variance discriminates four qualitatively different regions in the dynamical phase diagram that are indistinguishable by just looking at the mean magnetization. Second, bipartite entanglement between disjoint groups of the spins in the fully connected Ising model is studied. In the large N limit the entanglement Hamiltonian is shown to be a harmonic oscillator. A quantitative relation between the angular frequency of the oscillator and a squeezing measure of the spin state is derived. Explicit expressions for the Renyi entanglement entropies are given. In contrast to the variance, which vanishes in the thermodynamic limit, the entanglement entropy saturate to an N-independent positive value. The dynamics of the entanglement Hamiltonian and the entanglement entropies is discussed. Linear growth as a function of time is linked to the unstable fixed point and the homoclinic orbit of a pitchfork bifurcation, while logarithmic increase is explained due to a dephasing mechanism of nearby effective orbits. A generalization to other models that permit a semiclassical effective description (on a possibly higher dimensional phase space) is given. As a byproduct the connection between entangled Gaussian states and the symplectic eigenvalues of its reduced covariance is derived. An upper bound on the von Neumann entanglement entropy of any bipartite pure state is given in terms of the symplectic properties of its reduced covariance. Third, the existence of dynamical phase transitions (DPTs) in the fully connected Ising model is argued by two different approaches. DPTs are defined as non-analyticities in the time-dependence of the return probability rate function in the thermodynamic limit. Like entanglement entropy, the behavior of the return rate function is an observable-independent quantity used to characterize non-equilibrium states. A numerically based connection between the DPT and the analytic property of the infinite time averaged order parameter as a function of quench strength is discussed. The concept of Feynman history states is used to devise the notion of history entanglement. An information theoretic interpretation of the history entanglement entropy as the precision of a quantum clock follows from Holevo’s bound. The dependence of this entanglement entropy on the observation time constitutes an operator-independent characterization of the quantum dynamics in closed systems. For observation times comparable to the Heisenberg time, the history entanglement becomes sensitive to the statistics of the energy gaps, and thus distinguishes integrable and non-integrable dynamics. This is demonstrated for two examples, a random matrix ensemble, and a one-dimensional hardcore boson model with integrability breaking next nearest neighbor interaction.