Phase diagram for quantum Hall states in graphene

TL;DR

This paper studies quantum Hall states in graphene using Hartree-Fock approximation, revealing minimal Landau level mixing at various fields and showing stripe amplitudes scale with magnetic field, affecting phase stability.

Contribution

It demonstrates that Landau level mixing decreases with decreasing magnetic field in graphene, and shows stripe amplitudes scale linearly with magnetic field, impacting the phase diagram.

Findings

Minimal Landau level mixing at low magnetic fields.

Stripe amplitudes scale roughly linearly with magnetic field.

Multiple phases can persist down to zero magnetic field.

Abstract

We investigate integer and half-integer filling states (uniform and unidimensional stripe states respectively) for graphene using the Hartree-Fock approximation. For fixed filling factor, the ratio between the scales of the Coulomb interaction and Landau level spacing $g=(e^2/\epsilon \ell)/(\hbar v_F/\ell)$, with $\ell$ the magnetic length, is a field-independent constant. However, when $B$ decreases, the number of filled negative Landau levels increases, which surprisingly turns out to decrease the amount of Landau level mixing. The resulting states at fixed filling factor $\nu$ (for $\nu$ not too big) have very little Landau level mixing even at arbitrarily weak magnetic fields. Thus in the density-field phase diagram, many different phases may persist down to the origin, in contrast to the more standard two dimensional electron gas, in which the origin is surrounded by Wigner crystal states. We demonstrate that the stripe amplitudes scale roughly as $B$, so that the density waves ``evaporate'' continuously as $B\to 0$. Tight-binding calculations give the same scaling for stripe amplitude and demonstrate that the effect is not an artifact of the cutoff procedure used in the continuum calculations.