Topological insulators, spin, and the tight-binding method

TL;DR

This paper uses a symmetry-based tight-binding approach to identify the origin of the spin-dependent gap in graphene, clarifying the role of spin-orbit interactions and correcting existing models for the quantum spin Hall effect.

Contribution

It introduces a comprehensive tight-binding formulation that includes all symmetry-allowed interactions, providing deeper insight into topological states in graphene.

Findings

Identifies three-center interactions caused by spin-orbit coupling as the origin of the gap.

Provides a corrected Haldane model incorporating spin degrees of freedom.

Reproduces key features of the quantum spin Hall effect in graphene.

Abstract

As one of the first proposed topologically protected states, the quantum spin Hall effect in graphene relies critically on the existence of a spin-dependent gap at the K/K' points of the Brillouin zone. Using a tight-binding formulation based on the method of invariants, we identify the origin of such an intrinsic gap as the three-center interaction between the pi-orbitals caused by spin-orbit interactions. This methodology incorporates all symmetry compliant interactions previously neglected and has wider applications for comparisons between first-principle calculations and the tight-binding method. It also identifies a correction to the Haldane model and its generalization, which incorporates the spin degrees of freedom and reproduces all the salient features required for the quantum spin Hall effect in graphene.