Graphene and non-Abelian quantization

TL;DR

This paper introduces a nonrelativistic model based on deformed Heisenberg algebra to describe low energy excitations in graphene, reproducing key features like the Dirac spectrum and quantum Hall effect, with insights into interactions and external fields.

Contribution

It presents a novel nonrelativistic deformation of the Heisenberg algebra that captures graphene's low energy physics and reproduces its quantum Hall behavior.

Findings

Model reproduces the Dirac-like dispersion relation for graphene.

Hall conductivity aligns with the anomalous integer quantum Hall effect.

Interactions at first order do not alter the spectrum significantly.

Abstract

In this article we employ a simple nonrelativistic model to describe the low energy excitation of graphene. The model is based on a deformation of the Heisenberg algebra which makes the commutator of momenta proportional to the pseudo-spin. We solve the Landau problem for the resulting Hamiltonian which reduces, in the large mass limit while keeping fixed the Fermi velocity, to the usual linear one employed to describe these excitations as massless Dirac fermions. This model, extended to negative mass, allows to reproduce the leading terms in the low energy expansion of the dispersion relation for both nearest and next-to-nearest neighbor interactions. Taking into account the contributions of both Dirac points, the resulting Hall conductivity, evaluated with a $\zeta$-function approach, is consistent with the anomalous integer quantum Hall effect found in graphene. Moreover, when considered in first order perturbation theory, it is shown that the next-to-leading term in the interaction between nearest neighbor produces no modifications in the spectrum of the model while an electric field perpendicular to the magnetic field produces just a rigid shift of this spectrum. PACS: 03.65.-w, 81.05.ue, 73.43.-f