Exploring High-Order Topological States

In recent years,high-order topological states as a new topological quantum state have aroused extensive research interest and become a focus of theoretical and experimental research.Their discoveries have greatly enriched the types of topological states.Traditional body-edge corresponding is not available for higher-order topological states.High-order topological states have lower dimensional boundary states than traditional first-order topological states in the same dimension.For example,two-dimensional first-order topological states have one-dimensional gapless edge states,while two-dimensional second-order topological states have corner states.Not only high-order topological insulator but also high-order topological superconductor have caused widely research.Previous studies have shown that the hopping amplitudes in the unit cell is much smaller than that between the nearest-neighbor unit cells.Recently,by numerical calculation,we find that vortex can induce zero energy mode in first-order topological states and there has quantum tunneling effect between two vortices,the quantum tunneling strength is different with the different distance of two fluxs.Therefore,we propose a plan: considering a vortex superlattice based on the Chern insulator and P wave superconductor,where periodic arrangement of fluxes with different distance.The quantum tunneling strength between the vortex-induced states in the unit cell is much smaller than that between nearest-neighbor unit cells,then high-order topological states can be induced by the vortex superlattice.Firstly,two vortex defects on the Chern insulator can induce zero modes with local bound states.Due to two fluxes nearby,the inter-flux quantum tunneling effect will occur and the two zero modes couple.The energy splitting changes with different distance of two flux.Then,we consider a vortex superlattice based on the Chern insulator,where periodic arrangement of fluxes with different distance,like 4a and 7a.Here one unit cell consists of four lattice sites for vortex superlattice and the quantum tunneling strength between the vortex-induced states in the unit cell is much smaller than that between nearest-neighbor unit cells.By numerical calculation,we get all the eigenvalues and their wave functions,and find there are four zero-energy modes localized at four corners of the vortex superlattice,and the edge state is gapped,these all indicate the second-order topological states emerge from Chern insulator with vortex superlattice.After that,we also calculated the edge polarization to further verify its higher-order topological properties.Secondly,we use the similar methods to study vortex effect on P wave superconductor,and find periodic arrangement of fluxes with different distance can also induce zero mode with corner states.then,we also calculated the edge polarization to further verify its higher-order topological properties.Our work not only enriched the topological states but also provided a new insight for the realization of higher-order topological states.