Research On Topological Properties Of High-Order Topological Insulators

Searching for and researching novel state has always been one of the core directions of condensed matter physics.Since its appearance,topological state has aroused extensive attention and research in physics.In recent years,the emergence of high-order topological state concept further promotes the development of topological state.Compared with topological state,the"high order"of high-order topological state is embodied in its"bulk-edge correspondence",and the bulk-edge correspondence is a common feature of topological state.If the bulk property of a system is topology nontrivial,there will be stable and gapless edge states on its boundary accordingly.These edge states support the transport without dissipation,and have a broad application prospect in the manufacture of non dissipative electronic devices.For the traditional d-dimensional topological state,the non dissipative edge state appears on the d-1-dimensional boundary.For example,two-dimensional topological insulator has spiral edge state on its one-dimensional boundary,while three-dimensional topological insulator has Dirac surface state on its two-dimensional boundary.For higher order topological state,such as n-order topological state,its non dissipative edge state appears on the boundary of D-N dimension.For example,the edge state of the second-order two-dimensional topological insulator is in the corner of 0 dimension,and the edge state of the second-order three-dimensional topological insulator is on the edge of 1 dimension.Therefore,the topological insulator we mentioned in the past can be called first-order topological insulator.The above introduction reveals that the high-order topological insulator has new bulk-edge correspondence,which can make the non dissipative edge state more widespread.Therefore,the high-order topological insulator has attracted great attention since its appearance.The experimental and theoretical aspects have been studied rapidly,such as the materials to realize the high-order topology insulator,the stability of topological invariants,the symmetry classification of high-order topology and the lattice structure of the high order topology are still in the primary stage.This paper focuses on the stability of higher-order topology.In 2016,benalcazar,bernevig and Hughes extended the theory of electric dipole moment to high dimensions and proposed the first high-order topological insulator--quantum electric multipole moment insulator.These insulators are topological crystal insulators in nature,and they are protected by potential crystal symmetry,not nonspatial symmetry.For example,in the BBH model,topological properties are protected by mirror symmetries.In addition to mirror symmetries and other spatial symmetries,there are many irrelevant non spatial symmetries in the BBH model.We are very interested in the impact of these symmetry breaking on the topological quadrupole phase.Generally speaking,only mirror symmetries breaking will have an impact,while other symmetries breaking will not,but the fact is far from so simple.In addition,in this model,the topological index is constructed by the nested Wilson loop method,which is based on the topological equivalence between the Wannier band and the boundary energy spectrum,but this equivalence will disappear in many cases,so it is necessary to find a more effective topological index.By constructing several extended BBH models,we mainly summarize the following points:1.Chiral symmetry breaking will destroy the quadrupole phase and make the system enter the"indirect gap"phase,and this change can not be captured by the topological index constructed by the nested Wilson loop method.2.When the symmetry of time reversal is broken,the topological equivalence between the Wannier band and the boundary energy spectrum of the system is lost,that is,the nested Wilson loop method fails.In addition,the symmetry breaking of time inversion will lead to the topological phase transition from non trivial to trivial.3.The mirror symmetries breaking will not destroy the quantum electric quadrupole moment phase,and the electric quadrupole moment of the system remains quantized,which is different from the original BBH model.4.The electric quadrupole moment index obtained by the real space numerical calculation method is more effective than the nested Wilson loop method.5.The BBH system can induce higher-order topological phase under the a certain disorderV(R)(?)₄.