Interacting quantum Hall states in a finite graphene flake and at finite temperature

Abstract

The integer quantum Hall states at fillings $ν= 0$ and $|ν| = 1$ in monolayer graphene have drawn much attention as they are generated by electron-electron interactions. Here we explore aspects of the $ν= 0$ and $|ν| = 1$ quantum Hall states relevant for experimental samples. In particular, we study the effects of finite extent and finite temperature on the $ν= 0$ state and finite temperature for the $ν= 1$ state. For the $ν= 0$ state we consider the situation in which the bulk is a canted antiferromagnet and use parameters consistent with measurements of the bulk gap to study the edge states in tilted magnetic fields in order to compare with experiment [A. F. Young et al., Nature 505, 528 (2014)]. When spatial modulation of the order parameters is taken into account, we find that for graphene placed on boron nitride, the gap at the edge closes for magnetic fields comparable to those in experiment, giving rise to edge conduction with $G \sim 2e^2/h$ while the bulk gap remains almost unchanged. We also study the transition into the ordered state at finite temperature and field. We determine the scaling of critical temperatures as a function of magnetic field, $B$, and distance to the zero field critical point and find sublinear scaling with magnetic field for weak and intermediate strength interactions, and $\sqrt{B}$ scaling at the coupling associated with the zero field quantum critical point. We also predict that critical temperatures for $ν= 0$ states should be an order of magnitude higher than those for $|ν| = 1$ states, consistent with the fact that the low temperature gap for $ν= 0$ is roughly an order of magnitude larger than that for $|ν| = 1$.