Tutorial: Dirac Equation Perspective on Higher-Order Topological Insulators

TL;DR

This tutorial reviews how the Dirac equation framework describes higher-order topological insulators, highlighting their boundary states and symmetry protections in a pedagogical, self-contained manner.

Contribution

It provides a unified Dirac equation approach to understand higher-order topological insulators and their boundary states, connecting classical models with modern topological concepts.

Findings

Derivation of boundary and corner states using Dirac theory

Recasting SSH chain as a symmetry-protected boundary of 2D insulators

Analytical tractability of higher-order topological features

Abstract

In this tutorial, we pedagogically review recent developments in the field of non-interacting fermionic phases of matter, focussing 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 insulators 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.