Symmetry and Dirac points in graphene spectrum

TL;DR

This paper investigates how various symmetries in a hexagonal lattice influence the existence and stability of Dirac points in the spectrum of related operators, using symmetry-based mathematical proofs.

Contribution

It provides a comprehensive symmetry-based analysis of Dirac point existence and stability under different symmetry conditions in graphene-like structures.

Findings

Dirac points exist under certain symmetries like rotation and inversion.

Conical dispersion relations are stable under rotation symmetry.

Weak symmetry breaking can preserve Dirac points using geometric phase or eigenfunction parity.

Abstract

Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by $2\pi/3$ and inversion, rotation by $2\pi/3$ and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by $2\pi/3$ and vertical reflection. All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by $2\pi/3$. Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.