The Study On The Realization Scheme Of Higher-order Topological Corner States

The discovery of topological phases breaks the traditional paradigm of spontaneous symmetry breaking describing condensed matter phases,which provides a theoretical basis for understanding many novel physical effects.As a new class of topological phases,High-order topological insulators have greatly sparkled the interest of researchers.The discovery of higher-order topological phases enriches the types of topological phases of matter.Higher-order topological phases have lower-dimensional topological boundary states,that is,the boundary states of higher-order topological insulators of the same dimension have lower dimensions than that of traditional topological insulators with the same dimension.For example,two-dimensional second-order topological insulators have corner states protected by the topology.In order to better understand the properties of higher-order topological states,this paper propose two specific schemes for the realization of corner states of higher-order topological insulators,namely,the defect superlattice model and the anisotropic photonic graphene model.The properties of higher-order topological corner states are studied by analytical and numerical methods,including :(1)When a point defect is added to the Chern insulator with particle-hole symmetry,local bound states will appear.If regularly arranged defect superlattices with two different configurations are added into the Chern insulator,defect superlattices will exhibit different properties: Firstly,when the defects are arranged regularly,the superlattice induced localized states have similar topological properties.Secondly,when the defects are spaced alternatively,defects induced localized states exhibit properties of higher-order topological superconductors.Through specific research it is found that the alternative arrangement of defects gives rise to the nearest neighbor hopping having strong-weak alternating distribution,which leads to defect superlattice induced localized states having nontrivial bulk topological properties that can be characterized by edge polarizations and nested Wilson-loop.By means of numerically calculating energy level and particle-density distributions of the system with defect superlattices,we find the corners of defect superlattices host topologically protected localized corner states,which further confirms the defect superlattice induced bound states have topological properties of second-order topological insulator in two dimensions.(2)The corner states of higher-order topological insulators can be realized not only in Hermitian systems but also in non-Hermitian systems.In the non-Hermitian system of photonic graphene,photonic crystals provide a controllable platform for studying new topological states in open systems.In mirror-symmetric photonic graphene,topological corner states are induced by adjusting the gain and loss in the non-Hermitian system.The real and imaginary parts of the energy of photon graphene are related to the photon frequency and the gain and loss of photon.After the nontrivial topological corner states are calculated,we further calculate the topological invariants of the system by the Wilson-loop operator,and verify that the topological corner states are stable in the presence of mirror symmetric perturbations.Finally,the numerical results show that when the system has domain walls,there are topologically protected interface states localized at the domain walls,and the energy of the all states changes under the mirror symmetric perturbation,but the photonic topological interface states with purely imaginary energies still remain.