The Nonequilibrium And Entangment Properties Of One-dimentional XXZ Model

The quantum entanglement in the nonequilibrium dynamics of strongly correlated systems has been an increasingly active subject in condensed matter community.Entanglement is a key feature of quantum mechanics that makes it essentially different from the classical mechanics.After being proposed in 1930 s,it has attracted considerable attention and discussions,and is getting more and more popular.At present,people try to comprehend basic properties of spacetime and matter in a very fundamental level from the point of view of entanglement.In the fast-paced developing of quantum computation and quantum information,the concept of entanglement plays a key role.In condensed matter,people explore the properties of many-body systems from the perspective of entanglement,including the studies of strongly correlated phenomena,quantum phase transition,and classification of states of matter etc.In the past decades,thanks to the development of the significant progress in cold atoms,optical lattice and other highly controllable quantum simulating systems,one can re-produce and manipulate some amount of classical models in strongly correlated systems with high precision.This provides a new and prolific approach to the nonequilibrium properties of these systems,which also inspires considerable and intensive theoretical efforts on related issues.In this thesis we address the entanglement properties of the prestigious S = 1/2Heisenberg XXZ model in the quantum quench and the adiabatic process numerically via the exact diagonalization.The entanglement spectrum,entanglement gap and entanglement entropy are calculated.We verify and generalize the correspondence between the Loschmidt echo,a measurable quantity in nonequilibrium experiments,and the eigenvalues in the entanglement spectrum.A symmetric pattern between the overlap of the time-evolving wavefunction with the initial state,and with the vacuum state,during the 2? ? ? quench in the gapless phase,is identified.We also discuss the situation when the quench crosses the quantum critical point,and we find that the correspondence verified in the gapless phase can be extended to some extent beyond the critical points.The thesis is organized as follows: we first make a brief introduction to the background of the study in Chapter I,including quantum entanglement,quantum phase transition and the concept of nonequilibrium.In chapter II,the method of the exact diagonalization is detailed,exampled by the S = 1/2 XXZ Heisenberg model.There we demonstrate how to construct the basis and the Hamiltonian matrix,and how to partition a matrix into smaller blocks by considering the symmetry and good quantum numbers of the system,in order to enhance the numerical capacity and efficiency.After introducing the Lanczos algorithm in the Krylov space,we present its application in the matrix diagonalization and the time-evolving simulation.Chapter III is model-related,where we discuss the general properties of XXZ one-dimensional model and its phase diagram at zero temperature.The concept of the Tomonaga-Luttinger liquid and the technique of the phenomenological bosonization are explained.Our main results are summarized in Chapter IV,where the time-evolving wavefunction of the Luttinger liquid model after quench is obtained analytically.Based on the information of the wavefunction,the Loschmidt echo and the entanglement spectrum in the momentum space is calculated accordingly.The correspondence between these two kinds of quantities are confirmed and generalized.As a demonstration of the validity of the analytical results from the Luttinger liquid theory,we numerically investigate the quench process of the S = 1/2XXZ model.We make some discussions on the quench which in the gapless region and crosses the quantum critical point,thus we get some interesting conclusions.The entanglement gap and entangment entropy are calculated and discussed.The summary and outlook are given in chapter V.