Phase diagram of the ?=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 < v = 2 suggests, among other things, a quantum phase transition between valleyunpolarized and polarized states at a perpendicular electric-field D*. We find that the behavior of D* with v changes markedly as B is reduced. At v = 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 v = 2 state as a function of the magnetic and electric fields and predict that valley coherence may emerge for D & SIM; D* in the high-B regime. This suggests the system supports Kekule bond ordering, which could, in principle, be verified via scanning tunneling measurements.