Effective Hamiltonian for protected edge states in graphene

TL;DR

This paper introduces an effective two-band Hamiltonian for graphene that accurately describes protected edge states, highlighting band inversion as the key mechanism for edge state formation without relying on the full Brillouin zone analysis.

Contribution

The paper presents a novel effective Hamiltonian for graphene edge states based on a Taylor expansion around the M point, simplifying the analysis of topological edge states.

Findings

The Hamiltonian accurately models edge states in zigzag and armchair ribbons.

Edge states are governed by band inversion in reciprocal and real space.

The approach allows selecting the topological invariant point without altering the bulk spectrum.

Abstract

Edge states in topological insulators (TIs) disperse symmetrically about one of the time-reversal invariant momenta $\Lambda$ in the Brillouin zone (BZ) with protected degeneracies at $\Lambda$. Commonly TIs are distinguished from trivial insulators by the values of one or multiple topological invariants that require an analysis of the bulk band structure across the BZ. We propose an effective two-band Hamiltonian for the electronic states in graphene based on a Taylor expansion of the tight-binding Hamiltonian about the time-reversal invariant $M$ point at the edge of the BZ. This Hamiltonian provides a faithful description of the protected edge states for both zigzag and armchair ribbons though the concept of a BZ is not part of such an effective model. We show that the edge states are determined by a band inversion in both reciprocal and real space, which allows one to select $\Lambda$ for the edge states without affecting the bulk spectrum.