The Theoretical Study Of Higher-order Topological Insulators And Superconductors

The topological phase of matter is an important area of research in the condensed matter today.In particular,topological insulators and superconductors have been broadly studied in the past two decades.In recent years,the concept of the topological phase of matters has been extended to higher-orders.In this paper,we will introduce three works in the field of higher-order topological states: the higher-order topological superconductors based on weak topological insulators;higher-order topological phases emerging from Su-SchriefferHeeger stacking;and the general construction of higher-order topological insulators and superconductors.The main research results are as follows:(1)Starting from weak topological insulators and considering the interplay between superconductors and magnetic fields,we find that the helical or chiral Majorana hinge modes and even corner modes can be realized in this system.We find that these higher-order topological superconductors can be attributed to their certain boundaries(surfaces or hinges),which naturally form topological superconductor in DIII or D symmetry class.Correspondingly,the Majorana hinge and corner states can be characterized by the boundary topological invariants.Moreover,some chiral hinge states in our model naturally form the Majorana interference loop,reflecting the exceptional property of higher-order topological superconductors.Our models not only can be realized in iron-based superconductors with topological bands but also point out the direction of detecting the high-order topological matters through interference measurement.(2)We develop a systematical approach of constructing and classifying the twodimensional second-order topological phase.Our approach is based on the direct construction of analytical solution of the corner zero states in a series of 2D four bands models that stack the two Su-Schrieffer-Heeger(SSH)models.Our approach not only gives the celebrated Benalcazar-Bernevig-Hughes and 2D SSH models but also reveals a novel model.Moreover,we establish a unified topological characterization for these three models.Our principle of obtaining corner zero states can be readily generalized to arbitrary dimension and superconducting systems.Thus,our work sheds new light on the theoretical understanding of the higher-order topological phase and provides a broad way of realizing higher-order topological insulators and superconductors.(3)The boundary first-order topological phases(e.g.,surface Chern insulators)naturally have a boundary state with more than one dimension lower.Therefore,the most direct way to construct higher-order topological insulators and superconductors is to construct boundary first-order topological insulators or superconductors.Based on this visualized principle,we systematically construct higher-order topological insulators and superconductors of arbitrary dimensions with arbitrary order.In our general construction,we give the general form of Hamiltonian of topological multipole moment insulators and prove the existence of boundary multipole moments.Moreover,the boundary states of the higher-order topological models in our constructions are strictly solvable,and we give the analytic wave functions of these boundary states.Our study provides theoretical guidance for the design of higher-order topological insulators and superconductors in real materials.