Quantum Entanglement And Fidelity In Condensed Matter Systems

Quantum entanglement and quantum fidelity are two important concepts in the quantum theory. In the field of quantum information and quantum com-putation, the two concepts play the key roles. And they are not new ones in quantum theory, some researches bout them should trace back to the beginning of the quantum theory. Quantum entanglement reflects the basic properties of the quantum world such as the quantum coherence, probability and nonlocality. The quantum fidelity was used as a measure of stability and also used to study ir-reversibility, especially it is very useful in investigating quantum chaos. Recently, the study of entanglement properties and fidelity behaviors in many-body sys-tems have attracted much attention. The Heisenberg chains are widely studied in the condensed matter field. In this thesis, we will investigate the entangle-ment properties in a series of Heisenberg chains, and also study the dynamical behavior of entanglement. On the other hand, we will use fidelity and the fidelity susceptibility to indicate quantum phase transition.This thesis comes in four parts. The first part is Chapter 1 where we sys-tematically introduce the concepts, such as some measure of entanglement and the definition of fidelity and fidelity susceptibility. In the part, we introduce the negativity in detail.The second part includes Chapter 2,3 and 4 where we discuss the entangle-ment properties in some Heisenberg chains by uss of concurrence and negativity. In Chapter 2, we consider the spin-1/2 dimerized system and the XY system in external magnetic field. For small sizes system, the analytical results of entangle-ment can be obtained. By numerical calculation, we study the mean concurrence versus different interaction strength. In the XY system, we consider the effects of the stagger external field on the entanglement. In Chapter 3, we study the entan-glement properties in the (1/2,1) mixed-spin chain. In this system the negativity is a very well entanglement measure. In small systems, we can give the analytical results, and in the system with even sites, we can obtain the relation between entanglement and the system energy, which is a general and important result. Thermal entanglement is considered and the threshold temperature is given. In Chapter 4, spin-1 systems are considered. In the case with next-nearest-neighbor (NNN) interaction, we numerically study the entanglement versus NNN interaci-ton, and also present how the thermal fluctuation destroy entanglement.The third part includes Chapter 5 and 6, where we study the dynamical behaviors of entanglement. In Chapter 5, we consider two spins coupled to a environment which is characterized by a pure dephasing model. We assume the initial state of the two spins is two kinds of bound entangled states. We also choose two models to simulate the environment:one is free bosons in heat bath, and the other is a system consists of lots uncorrelated spin halves in thermal balance. Negativity and realignment are all used to characterized the dynamical properties of entanglement. In Chapter 6, we choose the Ising spin chain in a transverse field to act the environment system, which can be exactly calculated. The analytical results of the decoherence factors are given, and based on which we can obtain the expression of entanglement decaying with time. We focus on the critical point, at which the entanglement decays monotonously with time. When the two spins initially starts from a werner state, the complete disentanglement happens.The forth part is the chapter 7. In this part we first give a new concept called operator fidelity, which is state independent. In the system of Ising spin chain in a transverse field, the susceptibility of the operator fidelity presents special behavior at the critical point, which means that the operator fidelity can be a indicator of quantum phase transition. In another example system, the spin-1/2 system with NNN interaction, we consider the odd sites cases and the susceptibility of the operator fidelity can indicate the critical point overcoming the degenerate of ground state. In the second section of this part, we consider anisotropy (1/2,1) mixed-spin chain which also has quantum criticality. The fidelity, entanglement and purity are studied in this system. The three quantum all present special properties at the critical point.