Localization And Topology Of Non-hermitian Quasiperiodic Lattice System

Anderson localization transition in one-dimensional Aubry-André-Harper(AAH)model and the mobility edges in some quasiperiodic systems with generalized dual transformations have been extensively studied.In recent years,the non-Hermitian system has attracted attention for some researchers because of the novel and unique physical properties that are different from the previous Hermitian system.These characteristics are different from the usual Hermitian systems include but are not limited to complex energy singularities,biorthogonal basis vectors,non-Hermitian skin effects,and so on.Although the research on many aspects of the non-Hermitian system has made some progress,the research on the one-dimensional non-Hermitian quasiperiodic lattice system has been ignored.This thesis mainly studies the Anderson localization and topological phase transitions in one-dimensional non-Hermitian quasiperiodic lattice and the properties of non-Hermitian mobility edges.First of all,in order to understand the properties of the mobility edge in the onedimensional non-Hermitian system,in the second chapter we introduce a one-dimensional generalized AAH model with Parity-Time(PT)symmetry.We found that the nonHermitian mobility edge in this system only depends on the real part of the energy spectrum.That is to say,if real part of energy spectrum is above the critical real energy,then the single particle eigenstate corresponding to the real part of the energy spectrum is a localized state.The real part of energy spectrum is below the critical real energy then the corresponding single particle eigenstate is the extended state.Since the system also has PT symmetry,it has a PT symmetry breaking transition.What is newer and strange is that the energy corresponding to the extended state in this system must be a real number,and the energy corresponding to the localized state must be an imaginary number.This means that the delocalized to localized phase transition of the one-dimensional PT symmetry generalized AAH model is consistent with the PT symmetry breaking transition,that is,they have the same transition point.In addition,we also calculated the average inverse participation ratio(〈IPR〉)and the average normalized participation rate〈NPR〉 and average energy gap ratio 〈r〉 to describe the extended phase,the intermediate phase with non-Hermitian mobility side,and the localized phase in the system.We also have studied the dynamic behavior of the Loschmidt Echo(LE)in the non-Hermitian system,which can also confirm the existence of the non-Hermitian mobility edge.Secondly,in order to study the localization and topological properties of onedimensional Kitaev chain with non-Hermitian periodic and quasiperiodic potentials.We introduced the Kitaev chain with non-Hermitian periodic potentials in the first part of Chapter 3.In the certain parameter area,the non-Hermitian system is in the topological superconductor phase,and Majorana zero modes will be localized at the ends of the open chain.In the second part,we studied the Kitaev chain with non-Hermitian quasiperiodic potential.We used the transfer matrix method to obtain the topological invariants of the system,and also calculated the Lyapunov exponent to determine the phase boundary.In addition,based on the Lyapunov exponent,we can also distinguish the extended phase from the localized phase.Furthermore,we find that the transition from topological trivial phase to topological superconducting phase and the transition from localized to delocalized are consistent for the system.Finally,mobility edges and reentrant localization in the non-Hermitian dimerized quasiperiodic lattice is studied.When the non-Hermitian quasiperiodic potential in the dimerized system is uniform,we only find that the system has three phases,the extended phase,the intermediate phase,and the localized phase.Moreover,the energy corresponding to the extended state in the system is a real number,and the energy corresponding to the localized state is a complex number,so its localization transition is consistent with the PT symmetry breaking transition.When the non-Hermitian quasiperiodic potentials in the dimerized system are staggered,we find that there is one extended phase,three intermediate phases,and three localized phases and three localization transitions.We confirmed that the system under certain non-Hermitian parameter will undergo three localization transitions,and there will be multiple reentrant localization phenomena.In addition,if the non-Hermitian parameter are very large,we also find that the system will only undergo a localization transition,so there is no reentrant localization phenomenon at this time.