Nontrivial surface topological physics from strong and weak topological insulators and superconductors

TL;DR

This paper explores the surface states of various topological insulators and superconductors, revealing conditions under which gapped surfaces can host nontrivial topological phases, generalizing known effects across all symmetry classes.

Contribution

It generalizes the classification of surface topological states to all symmetry classes using Clifford algebra representation theory and constructs minimal Dirac Hamiltonians.

Findings

Surface of strong 3D topological insulator exhibits half quantum Hall effect.

Surface of weak 3D topological insulator shows half quantum spin Hall effect.

Gapped surface states can have nontrivial topological phases depending on symmetry class.

Abstract

We investigate states on the surface of strong and weak topological insulators and superconductors that have been gapped by a symmetry breaking term. The surface of a strong 3D topological insulator gapped by a magnetic material is well known to possess a half quantum Hall effect. Furthermore, it has been known that the surface of a weak 3D topological insulator gapped by a charge density wave exhibits a half quantum spin Hall effect. To generalize these results to all Altland-Zirnbauer symmetry classes of topological insulators and superconductors, we reproduce the classification table for the ten symmetry classes by using the representation theory of Clifford algebras and construct minimal-size Dirac Hamiltonians. We find that if the surface dimension and symmetry class possesses a $\mathbb{Z}$ or $\mathbb{Z}_2$ topological invariant, then the resulting surface state with a gapped symmetry breaking term may have a nontrivial topological phase.