Quantum Hall criticality and localization in graphene with short-range impurities at the Dirac point

TL;DR

This study investigates how short-range impurities affect the quantum Hall criticality and localization in graphene at the Dirac point, revealing complex scaling behaviors and the impact of different impurity types.

Contribution

It provides a detailed numerical analysis of conductivity and localization in graphene with short-range impurities, highlighting the role of impurity type and magnetic field strength.

Findings

Conductivity remains at the ballistic value for strong magnetic fields.

Weak fields lead to localization or critical behavior depending on impurity type.

All key phenomena of Anderson localization and criticality are observed in a unified setup.

Abstract

We explore the longitudinal conductivity of graphene at the Dirac point in a strong magnetic field with two types of short-range scatterers: adatoms that mix the valleys and "scalar" impurities that do not mix them. A scattering theory for the Dirac equation is employed to express the conductance of a graphene sample as a function of impurity coordinates; an averaging over impurity positions is then performed numerically. The conductivity $\sigma$ is equal to the ballistic value $4e^2/\pi h$ for each disorder realization provided the number of flux quanta considerably exceeds the number of impurities. For weaker fields, the conductivity in the presence of scalar impurities scales to the quantum-Hall critical point with $\sigma \simeq 4 \times 0.4 e^2/h$ at half filling or to zero away from half filling due to the onset of Anderson localization. For adatoms, the localization behavior is obtained also at half filling due to splitting of the critical energy by intervalley scattering. Our results reveal a complex scaling flow governed by fixed points of different symmetry classes: remarkably, all key manifestations of Anderson localization and criticality in two dimensions are observed numerically in a single setup.