Dirac equation perspective on higher-order topological insulators

In this Tutorial, we pedagogically review recent developments in the field of non-interacting fermionic phases of matter, focusing on the low-energy description of higher-order topological insulators in terms of the Dirac equation. Our aim is to give a mostly self-contained treatment. After introducing the Dirac approximation of topological crystalline band structures, we use it to derive the anomalous end and corner states of first- and higher-order topological insulators in one and two spatial dimensions. In particular, we recast the classical derivation of domain wall bound states of the Su–Schrieffer–Heeger (SSH) chain in terms of crystalline symmetry. The edge of a two-dimensional higher-order topological insulator can then be viewed as a single crystalline symmetry-protected SSH chain, whose domain wall bound states become the corner states. We never explicitly solve for the full symmetric boundary of the two-dimensional system but instead argue by adiabatic continuity. Our approach captures all salient features of higher-order topology while remaining analytically tractable.