The Study On The Properties Of Topological Corner States In Superfluids

The discovery of the topological phase has broken the traditional Landau phase transition theory and provided the basis for the exploration of new novel states of matter.Higher-order topological superfluid and higher-order topological superconductor have attracted a great deal of attention from researchers for their novel topological manifestations at low-dimensional boundaries(including corners and edges)and their potential applications in topological quantum computation.Different from conventional topological superfluid and superconductor,which have lower boundary state dimensions,for example,a two-dimensional second-order topological superfluid is characterized by its possession of topologically protected corner states.In general,the existence of higher-order topological corner states in superconductivity or superfluidity usually depend on complex superconducting pairings or superconducting heterojunction structures,therefore,the study of superconducting and superfluidic systems with simple superconducting pairings carrying topological corner states is important for the exploration and achievement of higher-order topological states.In this paper,based on topological superfluid with simplest s-wave pairing,the physical mechanisms of the Majorana corner states as well as higher-order topological corner states in the Bogoliubov excitation spectrum are investigated as follows:(1)Based on quantum spin Hall insulators with onsite attractive interactions,the sublattice dependent in-plane Zeeman field(two sublattices with opposite Zeeman energy)is used to attract Majorana corner states.Firstly,we start from a quantum spin Hall insulator with attractive interactions and use mean-field theory to derive the system ground state as s-wave superfluid state.The calculation reveals that as the Zeeman energy gradually increases,the boundary state energy gap gradually decreases to close and then reopen when the system has periodic boundary conditions in x-direction and open boundary in y-direction;while the boundary state energy gap always remains open when the x-direction is open and the y-direction has periodic boundary conditions.Therefore,it can be inferred that the topological phase transition occurs in the x-direction boundary state,while it does not occur in the y-direction.Next,the emergence of a second-order topological superfluid is verified by numerical simulations and it is characterized by the Majorana edge polarization approach.Finally,to better understand the reason for the appearance of Majorana corner states,the effective low-energy edge theory is explored,and it is found that the interaction of having two sublattice positions with opposite in-plane Zeeman energy and s-wave pairing leads to the opposite sign of the Dirac mass on the adjacent edge,which in turn leads to the appearance of Majorana corner states.(2)Based on s-wave superfluid,higher-order topological Bogoliubov excitation corner states are induced by a local potential with mirror symmetry.Firstly,the one-dimensional superlattice model for s-wave pairing is introduced and it is found that the mirror-symmetric potential between sublattices can drive the generation of topological phases and that its Bogoliubov excitation states have topological properties.Secondly,a two-dimensional s-wave paired superfluid is established,and one of the chains is modeled to have a mirror-symmetric potential distribution.The study found that there are four non-zero energy modes between the third and fourth energy bands of the model.The particle density distribution of the model is localized to both sides of the chain and has mirror symmetry.Finally,a hexagonal graphene lattice model with mirror-symmetric potential is established.Under the mean-field approximation,the s-wave superfluid in a hexagonal graphene lattice is studied using the solved Bogoliubov-de Gennes(Bd G)equation and a self-consistent iterative method.When the system exists with mirror-symmetric onsite potential,its Bogoliubov excitation corner states are found to have the same higher-order topological properties.