The Stability Of Majorana Fermions In One-dimensional Topological Model

The non-equilibrium evolution in the isolated quantum system can be driven by the method of quantum quenching,and the relevant behaviors of each topological model near the quantum critical point after quantum transformation can be analyzed by the topological invariants.In this paper,by using the one-dimensional dimerized Kitaev model,the survival probability of the zero mode of Majorana at both ends of the chain under different quenching paths is discussed.Before that,we first give the topological phase diagram of the model by analyzing the Hamiltonian of the system,and then calculate the energy spectrum under the periodic boundary condition and the open boundary condition.The results show that using the Majorana zero modes can distinguish the different topological phase regions sensitively.Then,the parameter space of Hamiltonian is determined by topological phase diagram,and the initial and final states with different topological properties are selected.It is found that even if the quenching path does not pass through the phase boundary,the zero mode survival probability of Majorana can reach zero when the final Hamiltonian parameter value is extremely close to the phase boundary,which indicates that the edge state has been coupled with the environment and lost its coherence.When the final Hamiltonian parameter value falls on the phase boundary,the survival probability function shows regular oscillation,and the amplitude and period of oscillation are related to the distance between the initial state and the final state in the parameter space.The closer the distance is,the more regular the function oscillation is.With the increase of the distance,the oscillation will gradually lose regularity or even double the period.In a pure dephasing model,we study the effect of the initial state energy variance on the behavior of the Loschmidt echo(LE)in a short time.We find that LE is a Gaussian function in a short time,and its width is determined by the energy variance of the initial state of the interacting Hamiltonian.When the initial state is the eigenstate of the interacting Hamiltonian,it is a quartic decay function,and its width is determined by the energy variance of the initial state of the commutator between the interacting Hamiltonian and the environmental Hamiltonian.In addition,the Gaussian envelope in the temporal evolution of LE in the strong coupling regime is determined by the inband variance.We will also verify the above conclusion in XY spin model(as environment).