Phase diagram of the $\nu =2$ 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 $1<\nu \le 2$ suggests, among other things, a quantum phase transition between valley-unpolarized and polarized states at a perpendicular electric field $D^{\ast }$. We find the behavior of $D^{\ast }$ with $\nu$ changes markedly as $B$ is reduced. At $\nu =2$, $D^{\ast }$ 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 $\nu =2$ state as a function of the magnetic and electric fields, and predict that valley-coherence may emerge for $D\sim D^{\ast }$ in the high $B$ regime. This suggests the system supports Kekule bond-ordering, which could in principle be verified via STM measurements.