Phase diagram of the ν=2 quantum Hall state in bilayer graphene

Bilayer graphene exhibits a rich phase diagram in the quantum Hall regime, arising from a multitude of internal degrees of freedom, including spin, valley, and orbital indices. The variety of fractional quantum Hall states between filling factors $1<\ensuremath{\nu}\ensuremath{\le}2$ suggests, among other things, a quantum phase transition between valley-unpolarized and polarized states at a perpendicular electric-field ${D}^*$. We find that the behavior of ${D}^*$ with $\ensuremath{\nu}$ changes markedly as $B$ is reduced. At $\ensuremath{\nu}=2, {D}^*$ may even vanish when $B$ is sufficiently small. We present a theoretical model for lattice-scale interactions, which explains these observations; surprisingly, both repulsive and attractive components in the interactions are required. Within this model, we analyze the nature of the $\ensuremath{\nu}=2$ state as a function of the magnetic and electric fields and predict that valley coherence may emerge for $D\ensuremath{\sim}{D}^*$ in the high-$B$ regime. This suggests the system supports Kekul\'e bond ordering, which could, in principle, be verified via scanning tunneling measurements.