Periodic Landau gauge and Quantum Hall effect in twisted bilayer graphene

TL;DR

This paper introduces a periodic Landau gauge approach to study the Hofstadter butterfly diagram and quantum Hall effect in twisted bilayer graphene, enabling analysis of magnetic flux effects in large supercells.

Contribution

It proposes a new periodic Landau gauge method that allows for the calculation of Hofstadter diagrams in twisted bilayer graphene with large supercells.

Findings

Periodic Landau gauge enables Hofstadter butterfly calculation.

Fractional magnetic flux through supercells is possible.

Quantized Hall conductance relates to Diophantine equations.

Abstract

Energy versus magnetic field (Hofstadter butterfly diagram) in twisted bilayer graphene is studied theoretically. If we take the usual Landau gauge, we cannot take a finite periodicity even when the magnetic flux through a supercell is a rational number. We show that the \textit{periodic} Landau gauge, which has the periodicity in one direction, makes it possible to obtain the Hofstadter butterfly diagram. Since a supercell can be large, magnetic flux through a supercell normalized by the flux quantum can be a fractional number with a small denominator, even when a magnetic field is not extremely strong. As a result, quantized Hall conductance can be a solution of nontrivial Diophantine equation.