Dirac Theory of Higher-Order Topology

The higher-order topological phases manifest a refined bulk-boundary correspondence ensuring gapless modes residing on lower-dimensional boundaries, such as corners and hinges, that remains operative across all the members of the topological family, namely insulators, superconductors and semimetals. In this talk, I will show that a hierarchy of topological phases can be constructed by combining crystallographic symmetry with the abstract mathematical representation theory of anticommuting Dirac matrices. Namely, an nth order topological phase with n>1 can be realized from its conventional or first-order counterpart (n=1) in the presence of n-number of suitable discrete symmetry breaking Wilson-Dirac masses. The resulting higher-order topological phase then supports gapless modes on boundaries, characterized by the codimension n. I will then pedagogically demonstrate its jurisdiction on a large variety of systems realized on a wide variety of material platforms. In particular, I will discuss a systematic construction of higher-order topological insulators, superconductors, Dirac and Weyl semimetals on regular crystals, amorphous networks, self-similar fractal lattices as well as periodically driven Floquet materials. The stability of higher-order topological phases against random charge impurities that are inevitably present in quantum materials will be briefly discussed as well. I will close this talk by pointing out the novel role of bulk crystal defects, such as dislocations and grain boundaries, as the smoking gun probe of higher-order topology.