Topological Property Of One-dimensional Non-hermitian Quantum System With PT Symmetry

Quantum mechanics describes the dynamics of microscopic particles,which is an important basis of modern physics.The observable quantities in quantum mechanics are represented by Hermitian operators,and the corresponding eigenvalues are real numbers.However,the eigenvalues of non-hermitian quantum systems with PT symmetry have also been proved to be real numbers within a certain range of system parameters.The verification of this theorem is presented by photonic experiment.In this paper,the basic concepts about quantum walk,the non-hermitian PT symmetric quantum theory and the topological characteristics of Hermitian AAH model are briefly introduced.On this basis,the effects of gain-loss terms on the topological properties of non-unitary quantum walk and non-hermitian AAH model with PT symmetry are investigated.We introduce two controllable parameters to the coin rotation operator and the conditional translation operator respectively,and add a gain-loss operator to each evolution step.A one-dimensional non-unitary quantum walk model with PT symmetry is structured.To characterize the topological characteristics of the system,the topological invariant — Zak phase is calculated and the topological phase diagram in adjustable parameter space is given.We also calculate the mean displacement,which is a experimentally observable quantity,to illustrate the topological phase transition of the system from the view of dynamics.Moreover,considering a boundary to divide the space into two parts with different topology,the evolution of the probability of the quantum walk on the boundary is studied.The existence of the topological protective bound state is verified.The non-hermitian AAH model with PT symmetry is constructed by introducing the gain-loss term into Hermitian AAH model.Its topological properties and the condition breaking PT symmetry are explored by observing the change of system eigenenergy with modulation phase andchemical potential strength.The global complex Berry phase as a topological invariant is calculated in the unbroken PT symmetry region.The influence of the next-nearest-neighboring intersite coupling is studied.It is found that it can lead to overlap between the two bands with the reason of breaking particle-hole symmetry.