Studies On The Kibble-Zurek Mechanism Of One Dimensional Localized And Critical Systems

In this paper,the Aubry-Andre-Harper(AAH)model of one-dimensional p wave superconductivity is studied,and two kinds of non-equilibrium dynamic behaviors are discussed.First,we analyze the phase transition classifications and universality class of the system,and then verify the non-equilibrium dynamic behavior of slow quench near the critical point.Secondly,we study the dynamics of sudden change of system parameters,and analytically study the system under two limiting cases of V=0 and V=?,which shows that the sudden quench between different phases will lead to the dynamical quantum phase transition.In addition,we systematically study the quench dynamics between these two limits by the numerical method.By studying the non-equilibrium dynamics of this system carefully,a different idea can be provided for the study of non-equilibrium dynamics in a one-dimensional quasi-periodic system.Firstly,in the one-dimensional generalized AAH model,we study the phase transi-tion properties and the slow quench dynamics of the system.The phase transition clas-sification of the system is obtained by the conventional ground state energy derivative and fidelity,and the dynamic critical exponent z and correlation length critical exponent v are given.By studying the dynamics of slow quench near the critical point,we found the scaling rate according to the Kibble-Zurek(KZ)mechanism hypothesis.In order to generalize the adiabatic-diabatic-adiabatic approximation,we calculate four different quenching ?(t)?t/?Q,-sign(t)| sin(t/?Q)|,-sign(t)| sin2(t/?Q)|,-sign(t)|t/?Q|1/2,where e(t)is a dimensionless constant,indicating the distance from the critical point,?Q with the quench time.Within the error,the scaling rate is basically the same as that of the KZ mechanism.We also supplement the size analysis of v and the critical exponents of other points on the localized-critical phase transition line.Except for the critical point?=0,the critical exponents of other phase transition points are basically consistent.Secondly,we study the sudden quench dynamics among localized phase,critical phase,and the extended phase.The zero point of Loschmidt echo can be regarded as the appearance of the dynamical quantum phase transition.In order to give an intuitive explanation of the zero point,we study the analytical solutions for two limiting sudden quench between V=0 and V=?.The results show that as long as the Hamiltonian before and after quench is not in the same phase,Loschmidt echo will reach zero point after some time intervals.Then,we calculate the quench process of arbitrary Vi and Vf between the two limits by the numerical method.Here Vi(Vf)represents the initial(final)value of the Hamiltonian.As long as the initial state and the final state are in the different phases,the dynamical quantum phase transition will occur.Moreover,when the initial state is in the critical phase and being quenched in different directions,the dynamical quantum phase transition behavior can be described by the above two limits.Therefore,Loschmidt echo can be used to describe the non-equilibrium dynamics of sudden quench between different phases.Finally,the summary and outlook of this paper are given.In this paper,we study the one-dimensional incommensurate AAH model with irrational number a in the po-tential energy,which will affect the universal class and scale behavior of the system.We can further study the non-equilibrium dynamics of the commensurate AAH model with rational number a,and try to answer the questions.For example,can it still be described by KZ mechanism and Loschmidt echo?In addition,when the system has the modulation on the potential energy and hopping term,it also has phase transition among localized phase,critical phase,and extended phase.So,is the non-equilibrium dynamics of this extended AAH model satisfying KZ mechanism scaling hypothesis?Is it still possible to describe the dynamical quantum phase transition by Loschmidt echo?All these are worthy of further studying.