Topological aspects of graphene: Dirac fermions and the bulk-edge correspondence in magnetic fields

TL;DR

This paper explores the topological electronic properties of graphene, demonstrating the robustness of Dirac fermions and quantum Hall effects across various lattice models, and establishing the bulk-edge correspondence in magnetic fields.

Contribution

It shows that Dirac fermions and the associated quantum Hall effects are topologically stable features of a broad class of two-dimensional lattices, not unique to honeycomb.

Findings

Dirac dispersions are generic in certain 2D lattices.

Quantum Hall effect persists over a wide chemical potential range.

Edge states in graphene are topologically protected and persist in magnetic fields.

Abstract

We discuss topological aspects of electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: (i) Appearance of the pairn of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is rather generic for a class of two-dimensional lattices that interpolate between square and $\pi$-flux lattices. Persistence of the peculiar QHE is interpreted as a topological stability. (ii) While we have the massless Dirac dispersion only around E=0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behavior acompanied by huge jumps in the QHE with a sign change. (iii) For edges we establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is a manifestation of the topological bulk-edge correspondence. We have also explicitly shown that the E=0 edge states in honeycomb in zero magnetic field persist in magnetic field.