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 $D^{*}$. We find that the behavior of $D^{*}$ with $\nu$ changes markedly as $B$ is reduced. At $\nu =2, D^{*}$ 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 $\nu =2$ 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 Kekulé bond ordering, which could, in principle, be verified via scanning tunneling measurements.

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• Landau levels
• Quantum Hall effect
• Topological materials
• Valley degrees of freedom
• Bilayer graphene
• Mean field theory
• Transport techniques