The Study Of "quenching" Dynamics In The Quasi-periodic Quantum Ising Model

Phase transition is a universal phenomenon in nature,according to the causes,we divide the phase transition into thermodynamic phase transition and quantum phase transition.Thermodynamic phase transition is caused by thermal fluctuation,and quantum phase transition is resulted from quantum fluctuation.System will produce quantum phase transition,when the parameter of the system passes through its critical point at absolute zero temperature.Once a quantum phase transition occurs in the system,the ground state of a many-body system will be changed.When the system is in the state with the energy gap,and the parameter of the system changes slowly enough,then,the system always evolves in its ground state,and we call this evolution adiabatic.However,the energy gap vanishes at the critical point.Consequently,no matter how slowly the parameter is driven the quantum state of the system cannot follow adiabatically the instantaneous ground state near the critical point.The excited state is a mosaic of ordered domains whose finite size depends on the rate of the transition.This scenario was first described by Kibble who pioneered the study of domain structure formation in the early universe and then its underlying dynamical mechanism was proposed by Zurek who predicted the domain wall density pk after evolution was related to transition time TQ.If the parameters of the system were changed by-t/?Q,Zurek shows ?? ? ?Q-? where ?= dv/(zv+1)is a combination of critical exponents.Here z and v are critical exponents,and d is the dimension of the system.Now we call the relationship between ?k and ?Q Kibble-Zurek mechanism.The essence of the mechanism is an adiabatic-impulse-adiabatic approximation.The main idea is that when a system is driven through a quantum phase transition,then the evolution of its state is adiabatic when the system is away from the critical point,but it must be nonadiabatic in a neighborhood of the critical point where the gap between the ground state and the first excited state tends to zero.The Kibble-Zurek mechanism have been verified in the thermodynamic system,at first.Recently,people begin to verify Kibble-Zurek mechanism in quantum systems.In the pure quantum Ising model,Dziarmaga came to the conclusion ??=1/2? 1/(?).This consistent with results predicted by KZM.In the random quantum Ising model,Dziarmaga shows ????Q-?.But ? vary with ?Q.Relatively fast quenches were fitted with the exponent ??0.48 which is close to the ??0.5 in the pure Ising model,for comparison,slowly quenches were also fitted by a power law????Q-?,but this time the exponent is a mere ??0.13.In this article,we first use of the numerical method to verify the above conclusions,we also study the relation between residual energy Ere and ?Q in pure and random system.Residual energy is the difference between the excited state energy and the ground state energy at the final time.We conclude that the greater the ?Q the smaller the residual energy.we study the relation between residual energy Ere and n which is the number of domain wall.In the pure model Ere and n meet a linear relationship,but in the random model Ere and n meet a nonlinear relationship.Our research is mainly focused on the quasi-periodic systems which is between random and period systems.The results showed in the first generalized Fibonacci quasi-periodic situation,the relation between ?? and ?Q is similar to pure Ising model.For the limited size of the system,there also will be a ?Qad.In evolution,When ?Q>?Q,ad,the evolution of the system can be approximated adiabatic,and ?Qad?N2.As can be seen from the residual energy,with the increase of ?q,Ere will be smaller.Then Ere and n meet a linear relationship which is similar to pure situation.In the second type of generalized Fibonacci quasi-periodic condition pk and ?Q were also fitted by ????Q-?.The curve has different powers,which is similar to the random situation.Relatively fast quenches were fitted with the exponent ??0.49 which is close to the ??0.5 in the pure Ising model,for comparison,slowly quenches were fitted with the exponent ??0.26.But there will be a ?Qad,and ?Qad?N2.This phenomenon is not seen in the random system.From the residual energy,the Ere will decrease with increasing?Q,and a nonlinear relationship with Ere and n.