Classification And Realization Of Topological Superconductors Protected By Spatial Symmetry

When the concepts of topology and homotopy equivalence were applied to the band theory,this ancient field underwent a revolutionary change.It quickly became apparent that gapped quantum many-body systems,which were considered equivalent from the viewpoint of Landau-Ginzburg theory,could in principle be topologically classified.Prior to this,it was generally assumed that all band insulators with the same symmetry group were essentially equivalent.Topological band theory predicts a class of extoic topological insulator phases that are characterized by gapless boundary states.Such phases have been experimentally discovered.As an conceptual analogue of the topological insulator,topological superconductors are a class of exotic topological phases with Majorana zero modes that can be used for fault-tolerant quantum computing.Therefore,finding systems that exhibit topological superconductivity has become a hot area of research in the past two decades.Among the many proposals for realizing topological superconductivity,they can be classified into intrinsic and extrinsic topological superconductors depending on whether the source of topology and superconductivity is the material itself or not.On the one hand,progress in the experimental search for intrinsic topological superconductors has been relatively slow,perhaps due to the rarity of unconventional superconductivity in nature.Another potential reason may be that the classification of intrinsic topological superconductors has not been established due to their unique complexity,which makes it unclear which crystal symmetries and pairing symmetries can give rise to nontrivial topological superconductors in experiments.In the first part of this paper,we develop a general method for obtaining surface states of higher-order topological phases using surface anomalies as the basis.This method cleverly avoids the complexity introduced by additional particle-hole symmetry relative to topological insulators,and constructively classifies all topological superconductors in DIII class with pairing symmetry under all crystal point group.On the other hand,to date,experimental schemes for realizing non-intrinsic topological superconductors can mainly be classified into two paradigms,the Fu-Kane vortex paradigm and the nanowire paradigm.However,these schemes have mutually restrictive experimental conditions,so exploring more experimental schemes for implementing topological superconducting platforms is also one of the main areas of research.In the second part of this paper,we demonstrate that it is possible to get superconductors with specific topology by artificially introducing specific magnetic orders based on band degeneracies at high-symmetry points in the Brillouin zone.The advantage of this scheme is that,as a topological superconductor designed by symmetry-protected band degeneracies,only the spatial group symmetry of the crystal lattice structure needs to be ensured in experiments.