Non-bloch Band Theory Of Non-hermitian Topological System

The bulk-boundary-correspondence is an essential principle for our research on topological states and topological phases.Generally speaking,a nontrivial bulk topology usually implies the emergence of edge states under the open boundary conduction,that is to say,the topological invariant obtained by the Bloch Hamiltonian defined on Brillouin zone can accurately predict the topological nontrivial edge states under the open boundary condition.Recently,with the continuous research of non-Hermitian quantum systems,more and more people have realized that traditional bulk-boundary-correspondence is no longer applicable in nonHermitian systems.For example,the discovery of the non-Hermitian skin effect,and the difference between the energy spectrum under open boundary conditions and periodic boundary conditions.These phenomena cannot be well explained by Bloch band theory.Further studies revealed that all the eigenstates localized the boundary which called the skin effect and the difference of energy spectrum under different boundary conditions often occur simultaneously.In order to restore the bulk-boundary-correspondence,professor Wangzhong of Tsinghua University extended the wave number k from the real number domain to the complex number domain,which is to extend the Brillouin zone to the generalized Brillouin zone.With the help of the generalized Brillouin zone,it is well explained that all eigenstates in the non-Hermitian system are localized to the boundary,and the huge difference in energy spectrum under different boundary conditions.At the same time,the non-Bloch topological invariant defined on the generalized Brillouin zone can also faithfully describe the existence of the corresponding topological nontrivial edge state.So far,the non-Bloch bulk-boundary-correspondence has been established.No matter it is a Hermitian system or a nonHermitian system,which can be matched very well.Based on the Non-Bloch band theory,this paper analyzes the differences in eigenvalues,eigenvectors,exceptional points,and complex energy gaps between non-Hermitian and Hermitian systems.We explain the relationship in non-Hermitian system among the skin effect,differernce of energy spectrum under different boundary condictions and generalized Brillouin zones by taking the one-dimensional non-Hermitian Su-Schrieffer-Heeger model as an example.Then,we introduce two methods for calculating the generalized Brillouin zone,the classification of symmetry in non-Hermitian system,and the influence of different symmetry on energy and generalized Brillouin zone.In one-dimensional non-Hermitian superconductors,we investigate the energy spectrum under different boundary conditions and the Z skin effect.According to the partice-hole symmetry,we found there exist reciprocal particle and hole loops of generalized Brillouin zone.The critical point of quantum phase transition,where the energy gap closes,appears when the particle and hole loops cross at Brillouin zone.Also,we interplay the non-Hermitian and the periodic driving in a one-dimensional Su-Schrieffer-Heeger model,a novel phenomenon can emerge: the robust edge states can appear even when the Floquet bands are topological trivial with zero non-Bloch band invariant,which is defifined in terms of the non-Bloch effective Hamiltonian.We also show the relation between the non-Bloch winding numbers and the non-Bloch band invariant.Then the bulk-boundary-correspondence in non-Hermitian periodically driven system is established.Finally,we use low-energy continuum case as the tool to obtain the topological phase diagram of the non-Hermitian Weyl semimetal,which is also confirmed by the energy spectra from our numerical results.Moreover,these Fermi-arc edge modes can manifest as the unidirectional edge motion,and their signatures are consistent with the non-Bloch bulk-boundary correspondence,which defined by the non-Bloch chern number.