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

The integer quantum Hall states at fillings nu = 0 and vertical bar nu vertical bar = 1 in monolayer graphene have drawn much attention as they are generated by electron-electron interactions. Here we explore aspects of the nu = 0 and vertical bar nu vertical bar = 1 quantum Hall states relevant for experimental samples. In particular, we study the effects of finite extent and finite temperature on the nu = 0 state and finite temperature for the nu = 1 state. For the nu = 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 (London) 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 similar to e(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 root B scaling at the coupling associated with the zero-field quantum critical point. We also predict that critical temperatures for nu = 0 states should be an order of magnitude higher than those for vertical bar nu vertical bar = 1 states, consistent with the fact that the low-temperature gap for nu = 0 is roughly an order of magnitude larger than that for vertical bar nu vertical bar = 1.