Theoretical Study On Two-Dimensional Second-Order Topological Systems

Condensed matter systems contain profound physics due to the diverse atomic and electronic structures and presents rich and colorful phases of materials.According to Landau’s theory of phase transitions,in matter,the change of the state reflects the evolution of some order,and the phase transition corresponds to the process of symmetry breaking.For a long time,it was thought that all phase transitions could be explained by Landau theory until topological phase transitions were proposed.BKT phase transition introduces the concept of topological order for the first time which does not involve the breaking of symmetry,so it cannot be explained by Landau phase transition theory.The discovery of the quantum Hall effect had brought attention to a special kind of two-dimensional material platform,which is insulating inside,but the carriers can be transmitted along the edge without backscattering.Quantum Hall effect requires external magnetic field to break the time reversal symmetry of the system.Later,the quantum spin Hall effect system with time reversal symmetry which does not need any magnetic field was proposed.This system supports a pair of spin-momentumlocked boundary states at the edge,which leads to the concept of topological insulator.After years of theoretical and experimental development,the research on traditional topological insulators has been gradually refined.In recent years,the concept of topological insulators has been extended to a variety of other platforms,such as systems of phonon,photons,acoustics,circuits,etc.Besides,the higher-order topological systems with lower dimensional topological local states have been developed.The ddimensional n-order(2≤n≤d)higher-order topological insulators support(d-n)dimensional local states.In two-dimensional systems,the higher-order topological insulators usually refer to second-order topological insulators and support 0dimensional topological corner states.In this thesis,several two-dimensional secondorder topological systems are studied,which will be divided into five chapters to report.In Chapter 1,the origin and development of topological concepts in condensed matter systems are introduced,including Landau theory of phase transitions,BKT phase transition,the quantum Hall effect,the quantum spin Hall effect,and the twodimensional higher-order topological insulators are introduced.Then it introduces the discovery of superconductivity,the BCS theory describing the microscopic mechanism of superconductivity,and the concept of topological superconductor and Majorana zero mode.In Chapter 2,some theories and calculation methods used in the research are introduced.The introductions are mainly from three aspects:firstly,the development and theoretical bases of the density functional theory;secondly,some common implementation methods for calculation of energy band;and finally,several calculation software used in the research of this thesis.In Chapter 3,a theoretical study of the coexistence of phononic and electronic second-order topological insulators in graphdiyne is introduced.Graphdiyne is a twodimensional planar carbon structure which is similar to graphene,but different intracell and inter-cell couplings open its energy gap.It has been reported that graphdiyne is an electronic second-order topological insulator.By model analysis and numerical calculations,we demonstrate that graphdiyne supports both out-and in-plane phononic second-order topology.We also calculated the electron-phonon couplings(EPCs)between the topological phononic and electronic corner states.The results reveal the enhanced EPCs or cancelled EPCs in graphdiyne which depend on the mirror symmetry of the phonon.In Chapter 4,the theoretical research on two second-order topological phenomena in the two-dimensional FeSe is introduced.The first is the second-order topological insulators induced by spin tilting.The FeSe monolayer with checkerboard antiferromagnetic configuration has topological helical edge states that can be gapped by the in-plane magnetic field.The edge state solutions of the effective model show the existence of fractional mass-kink in the rectangular clusters which will support the 0dimensional topological corner states.Then numerical calculations based on firstprinciples show the same results as the model.Another second-order topological phenomenon is the 0-dimensional topological defect states found experimentally on the two-dimensional surface of FeSe,which is reasonably explained by our theoretical calculations and analyses.The topological dislocation defects in geometry couple with the topological electronic states,and the mass domain walls generated at the center of the dislocation defects lead to the topological defect states.In Chapter 5,the theoretical research of topological dislocation states and secondorder topological superconducting states in the Bi monolayer is introduced.The Bi monolayer is a two-dimensional topological insulator,and topological defect states may occur when the electronic topology meets the topology of structure defects.It is found that 6-8 ring like dislocations support topological defect states,while 5-7 ring like dislocations do not.The results of calculations and analyses finally reveal the physical mechanism of whether topological defect states can be generated.The superconducting proximity effect can introduce superconductivity into the helical edge states of the Bi monolayer.Under an in-plane Zeeman field,the gap of edge states will undergo a topological phase transition.By taking advantage of the different responses of different edge terminations to magnetic field,the mass inversion can be generated at the intersection corner of two edges with different terminations,thus inducing a Majorana zero mode.In addition,it is found that the energy degeneracy of the same type of edge(e.g.zigzag)can be broken by applying uniaxial stress,so that the mass terms of edge states at the intersection corner of edges with the same type can be reversed to induce the Majorana zero mode under an appropriate in-plane Zeeman field.In Chapter 6,the total summary and some prospects of our research are made.