Several Quantum System Heating Problems And One-dimensional System PT Symmetry Breaking Transition Studies

The nonequilibrium dynamics of isolated many-body quantum systems has long been an elusive subject. Due to spectacular experimental progress in ultracold atomic gases in optical lattices and important developments in numerical methods that allow theoret-ical insight into many-body quantum systems, the nonequilibrium dynamics of isolated many-body quantum systems has drawn renewed attention recently. In this context, the topic of thermalization in isolated many-body quantum systems has received fresh and growing attention. The term thermalization is referred to a relaxation to states in which the values of observables are stationary, and predictable using conventional statistical ensembles. The study of thermalization in isolated many-body quantum systems is not only of fundamental importance for many different fields, ranging from quantum statisti-cal mechanics to mathematical physics, quantum chaos, transport problems, many-body localization, integrable and nonintegrable dynamics, but could also be crucial for future technologies, such as quantum computer and quantum communication.On the other hand, we have studied the spontaneous PT-symmetry breaking transi-tion (SPTBT) in non-Hermitian PT-symmetric Hamiltonian systems. In 1998, Ben-der and Boettcher found that a non-Hermitian Hamiltonian with PT symmetry can have a completely real spectrum. Since then, there has been growing interest in non-Hermitian PT-symmetric Hamiltonian systems. An extremely interesting property of a non-Hermitian PT-symmetric system is the transition from an unbroken PT-symmetry phase to a spontaneous-PT-symmetry-broken phase. In the unbroken PT-symmetry phase, the system has a completely real spectrum and all the eigenfunctions of the Hamil-tonian are also eigenfunctions of the PT operator, i.e., all the eigenfunctions have PT symmetry. However, in the spontaneous-PT-symmetry-broken phase the spectrum be-comes partially or completely complex and not all the the eigenfunctions of the Hamilto-nian are PT symmetric. In recent years, there has been growing interest in non-Hermitian PT-symmetric Hamiltonian systems. Examples of such systems range from quantum field theories to open quantum systems,the Anderson models, the tight binding chain,the spin chain and topological models, as well as the optical systems. Moreover, crucial advances in photonic lattices and photonic crystals have made it possible to experimentally imple-ment non-Hermitian PT-symmetric systems. The study of non-Hermitian PT-symmetric Hamiltonian systems is important for the development of quantum mechanics. Further-more, the development of the non-Hermitian theory and implementation of non-Hermitian systems make the realization of new optical and quantum devices possible.In this thesis, we will study the nonequilibrium dynamics and thermalization in iso-lated many-body quantum systems. We also investigate the spontaneous PT-symmetry breaking scenario in non-Hermitian PT-symmetric Hamiltonian systems.1. The nonequilibrium dynamics and thermalization in isolated many-body quantum systemsSo far much study on thermalization in isolated many-body quantum systems has been done for homogeneous systems, revealing that the behavior of observables after re-laxation is generally different for integrable and non-integrable homogeneous systems.In the case of homogeneous non-integrable systems, thermalization is expected and ob-servables relax to values obtained by conventional statistical ensembles. On the other hand, in homogeneous integrable systems, observables after relaxation can be described by generalized Gibbs ensembles (GGE) instead of conventional statistical ensembles, a process we call generalized thermalization. For isolated inhomogeneous many-body quan-tum systems, much study shows that many-body localization may occur in inhomogeneous non-integrable systems and it can lead to lack of thermalization after relaxation. On the other side, some study for inhomogeneous integrable systems shows that the fate of the GGE description is related to the properties of single-particle eigenstates of the system,for example, a study of hard-core bosons (HCBs) in the Aubry-Andre model shows that the GGE provides a good description of observables after relaxation in the delocalized regime in which single-particle eigenstates are extended, and that the GGE fails to de-scribe observables that depend on nonlocal correlations after relaxation in the localized regime in which the single-particle eigenstates are localized. Therefore, it is natural to ask: How is the fate of GGE description when the model has critical single-particle eigen-states that are neither localized nor extended? How is the applicability of the GGE when localized and extended single-particle eigenstates coexist? To address these questions,in this thesis we will study the nonequilibrium dynamics of HCBs in lattices with binary quasiperiodic potentials, in quasiperiodic optical lattice models with single-particle mo-bility edges and in driven lattices with constant force. Specifically, we consider systems in which the number of particles is half filling.Firstly, we study the nonequilibrium dynamics of HCBs in Fibonacci and generalized Fibonacci (GF) lattices following a sudden quench. The Fibonacci lattice is one of the simplest prototypes of quasiperiodic systems and it has been rigorously proved that the Fibonacci lattice has critical single-particle eigenstates, which means that the eigenstates are neither localized nor extended. The generalized Fibonacci lattice is another well-studied kind of quasiperiodic system and it can be classified into two classes according to whether or not the lattice possesses the Pisot-Vijayaraghavan (PV) property. The first class possesses the PV property and shows properties of single-particle eigenstates similar to those of the Fibonacci lattice. In contrast, the second class is known to possess no PV property, and there exist critical single-particle eigenstates, as well as localized and extended single-particle eigenstates. We find that in Fibonacci lattices the observables relax to time-independent values after quenches. It is found that after relaxation the GGE provides a good description for local observables, such as the on-site density, while GGE fails to describe observables that depend on nonlocal correlations, such as the momentum distribution function and the occupation of the natural orbitals. In addition, the grand-canonical ensemble fails to describe all observables after relaxation since these systems are integrable. These results are similar to those of HCBs in the localized regime of the Aubry-Andre model. We also studied the dynamics of HCBs in GF lattices and find that the relaxation dynamics and the resulting states after relaxation are similar to those in Fibonacci lattices.Then, we study the quench dynamics of HCBs in three quasiperiodic lattice models in the presence of single-particle mobility edges. For the first model, all single-particle eigenstates with energy |E| < Ec are extended and all others are localized. In the sec-ond model, there is a mobility edge separating localized and extended single-particle eigenstates at an energy Ec. For eigenenergy E < Ec, the corresponding single-particle eigenstates are localized. On the contrary, the single-particle eigenstates are extended when E > Ec. For the third model, all single-particle eigenstates with energy |E| > Ec are extended and all others are localized. We find that in the first model the observables relax to time-independent values after quenches and after relaxation the GGE provides a good description for the observable. These results are similar to those of HCBs in the delocalized regime of the Aubry-Andre model. However, in the second and third models,GGE fails to describe observables that depend on nonlocal correlations (such as the mo-mentum distribution function and the occupation of the natural orbitals), which is similar to that of HCBs in the localized regime of the Aubry-Andre model.Thirdly, we study the quench dynamics of HCBs in driven lattices with constant force.At the initial time, we consider a sudden quench in which an additional constant force F is abruptly introduced to a homogeneous lattice. In a homogeneous lattice, the single-particle eigenstates are extended, and when an additional constant force is introduced,the eigenstates become localized. We numerically calculate the time evolution of the on-site density, the momentum distribution function and the occupation of the natural orbitals following a sudden quench. We find that, after relaxation the GGE provides a good description for the on-site density, while the long-time averages for the momentum distribution function and the occupation of the natural orbitals are different from the predictions of the GGE for them. In addition, unlike HCBs in the other lattice models, the momentum distribution function can not relax to time-independent values after quenches.2. The spontaneous PT symmetry breaking transition in non-Hermitian PT-symmetric systemsA recent study of a non-Hermitian PT-symmetric topological model (SSH model)shows that the system undergoes a SPTBT in the topologically trivial phase (TTP)region, while in the topologically nontrivial phase (TNP) region there exists only a spontaneous-PT-symmetry-broken phase. Therefore, it is natural to ask a question: Is there an universal correlation between topological properties and the SPTBT? To ad-dress this question, we investigate the spontaneous PT-symmetry breaking scenario in the other two non-Hermitian PT-symmetric topological models. The first model we study is the non-Hermitian PT-symmetric Kitaev model. It is found that a SPTBT occurs in the TTP region, similar to that of the SSH model. However, unlike the SSH model,the system also undergoes such a transition in the TNP region. The second model is a non-Hermitian PT-symmetric extended Kitaev model. The extended Kitaev model is comprised of the Kitaev and SSH models and contains three different topological phases:the TTP, the Kitaev-like TNP and SSH-like TNP. We find that, a SPTBT occurs in the Kitaev-like TNP region, as well as in part of the TTP and SSH-like TNP regions, whereas there exists only a spontaneous-PT-symmetry-broken phase in the rest of the TTP and SSH-like TNP regions. Therefore,we can conclude that there is no universal correlation between topological properties and the SPTBT.