Research On Dynamical Quantum Phase Transitions And Related Non-equilibrium Dynamics

Quantum phase transitions have long been a hot spot in condensed matter physics,and the study of quantum phase transitions in non-equilibrium systems also gives us a deeper understanding of quantum physical properties.First,for the dynamical quantum phase transitions in the closed quantum system,the non-equilibrium phase transitions can be diagnosed by using the exact zero of the Loschmidt echo in thermodynamic limit or the long-time average of an observable.Second,the non-equilibrium phase transitions can be induced by the dissipation and measurement in the open quantum system.For the study of the dynamical quantum phase transitions in the closed quantum system,the discussion of the exact zero of the Loschmidt echo in previous works is often under the thermodynamic limit.In Chapter 2,we shall study exact zeros of Loschmidt echo for dynamical quantum phase transitions in finite size systems.Our results unveil that exact zeros of Loschmidt echo exist even in finite size quantum systems when the prequench parameter and the postquench parameter satisfy the constrain relation and take values from different phases.We further analyze the time for the appearance of the first exact zero of Loschmidt echo which is known as the quantum speed limit time.By analyzing the maximum quantum speed limit time,we find that when the system is quenched from a non-critical state to a critical phase,the quantum speed limit time will approach to infinity.We also calculate the minimal value of the quantum speed limit time and find that its behavior is dependent on the initial phase.On the other hand,characterizing the phase transition of a non-equilibrium system by the long-time average of an observable often depends on the choice of the observ-able.In Chapter 3,We shall unveil the role of the long-time average of the Loschmidt echo in the characterization of non-equilibrium quantum phase transitions.By study-ing sudden quench processes across quantum phase transitions in various quantum sys-tems,we demonstrate that non-equilibrium quantum phase transitions can be identified by nonanalyticities in the long-time average of the Loschmidt echo or the correspond-ing rate function or the emergence of divergence in the second derivative of the rate function when the driving quench parameter crosses the phase transition points.The connection between the second derivative of the rate function and fidelity susceptibility is also discussed,which means that using the long-time average of the Loschmidt echo to characterize the dynamical quantum phase transitions is universal.For an open quantum system that satisfies the Markov process,usually the dy-namic equation of the system satisfies the Lindblad master equation.The Liouvillian gap can be calculated from the Lindblad master equation,and the inverse of the Li-ouvillian gap can be used to describe the relaxation time of the system.However,the size-scaling relation of the Liouvillian gap is still not clear in the presence of Anderson localization.In Chapter 4,we shall study the non-equilibrium dynamics of Anderson localized systems driven by boundary dissipation,and unveil that the Liouvillian gap fulfills an exponential scaling relation Δ_g～e^-KL in the localized phase.By scruti-nizing the extended Aubry-André-Harper model,we fit the value of a and find it to be in good agreement with the analytical result of Lyapunov exponentwhich is the inverse of the localization length.We further apply the perturbation theory to derive Δ_g～e^-KL analytically.The exponential scaling relation was verified to hold true in other one-dimensional quasiperiodic models and random disorder models.We also ex-amine the relaxation time of average occupancy number operator and show the inverse of Liouvillian gap giving a reasonable time scale for the system achieving the steady state.