Dynamical Behaviors In The Vicinity Of The Quantum Phase Transitions And Their Semiclassical Studies

Presently, there are many concepts and quantities in the field of the quantum in-formation, such as, fidelity, Loschmidt echo(LE), entanglement and quantum discord, which are used to analyze properties of the critical points in the quantum phase tran-sitions (QPT). In many models, researchers found that these concepts and quantities have special properties such as fast decay or singularity, which can characterize the ap-pearance of the quantum phase transition. However, it’s hard to get the nature of these quantities from the view of the analytical analysis, because the analytical analysis needs the whole information of the ground state and excitation states with different parameter-s. Although there are some studies which use the perturbation theory or Kibble-Zurek mechanism to analyse some properties of fidelity or Loschmidt echo, they are limit-ed in one model or one class of models, and can’t give the whole properties of these quantities.In our research, we give an analytical method to study the dynamical behaviors in the vicinity of the critical points. In one class of QPT which has infinite degenerate of the ground states at the critical points, we prove the semi-classical theory can be used to analyse the dynamic property of the system in the nearby of the critical points, and discuss the condition for validity of this semi-classical theory near the critical points. These are rare in the previous researches. As an application, the prediction of the behav-iors of LE in different systems are given, too. Using the Van Vleck propagator which is obtained from the path integral semi-classical theory, one can easily obtain such LE behaviors. In the vicinity of the critical points, depending on the different dimension of the classic corresponding system, LE has two kinds of behaviors. Namely, in a one-dimensional classic corresponding regular system, LE has a Gaussian decay for initial times and a l/t decay for long time. When the dimension of the classic corresponding regular system is sufficiently large, the LE has the exponential decay.At present, there are many studies focus on the scaling behaviors of the quantities near the critical points, and try to figure out how these quantities depend on their con-trol parameters. In the previous studies, the characteristic energy scale usually takes the form Ec～|λ-λc|φ in the vicinity of the critical points, with Ac indicating the critical point and φ a critical exponent. However, these quantities studied are only the function of one parameter. There are two control parameters A and (?) for fidelity and LE. In this situation, how these quantities depend on their double control parameters? In2011, Rams and Damski give the scaling behaviors of fidelity in the Ising model, which depending on their double control parameters independently. Through the ana-lytical analysis, in a single bosonic zero mode QPT, we found a special scaling behavior for fidelity. This new scaling behavior only depends on the ratio η of the parameters (?) and A’, with η=(?)/(?). This scaling property is valid also for the time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.Next, in the vicinity of the critical point of the single mode Dicke model and Ising model, we validate the predictions of the semi-classical theory for LE and the new found scaling behavior. These predictions fit well with the numerical calculation. In addition to this, we also discuss the validity of the semi-classical prediction in the deep quantum region of the Ising model and give the change from validity to breakdown. In one quantum chaotic model such as the sawtooth model, we discuss this validity too. In our study, we found the semi-classical predictions for the LE work well even in the deep quantum region, especially in the FGR regime of the quantum chaotic systems.