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<ν≤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 ν changes markedly as B is reduced. At ν=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 ν=2 state as a function of the magnetic and electric fields and predict that valley coherence may emerge for D∼D∗ in the high-B regime. This suggests the system supports Kekulé bond ordering, which could, in principle, be verified via scanning tunneling measurements.