Quantum Hall effect in graphene: a functional determinant approach

C G Beneventano,E M Santangelo

doi:10.1088/1751-8113/41/16/164035

Abstract

We start the paper with a brief presentation of the main characteristics of graphene, and of the Dirac theory of massless fermions in 2+1 dimensions obtained as the associated low-momentum effective theory, in the absence of external fields. We then summarize the main steps needed to obtain the Hall conductivity in the effective theory at finite temperature and density, with emphasis on its dependence on the phase of the Dirac determinant selected during the evaluation of the effective action. Finally, we discuss the behavior, under gauge transformations, of the contribution due to the lowest Landau level, and interpret gauge transformations as rotations of the corresponding spinors around the magnetic field.

Highlights

Graphene is a bidimensional array of carbon atoms, packed in a honeycomb crystal structure
The first conclusion, as already stated in [6], is that two of the three possible combinations of phases give behaviors of the Hall conductivity which coincide with those measured in monoand bilayer graphene
In the case of bilayer graphene [3], the Hall conductivity presents a jump of height 2 for νC = 0, and further jumps of height 1

Summary

Introduction

Graphene is a bidimensional array of carbon atoms, packed in a honeycomb crystal structure. In [5], we showed that a field theory calculation at finite temperature and density, based upon ζ -function regularization of the Dirac determinant leads, in the zero-temperature limit, to a sequence of plateaux in the Hall conductivity consistent with the measured ones, each time the chemical potential goes through a nonzero Landau level. It was shown in [6] that two of the three possible combinations of phases of the Dirac determinant in both nonequivalent Clifford representations predict a behavior around zero chemical potential consistent with the ones measured in mono- and bi-layer graphene.

Structure of graphene: effective continuous model

Conclusions