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Bilayer graphene (BLG) offers a rich platform for broken symmetry statesstabilized by interactions. In this work we study the phase diagram of BLG inthe quantum Hall regime at filling factor $\nu=0$ within the Hartree-Fockapproximation. In the simplest non-interacting situation this system has eight(nearly) degenerate Landau levels near the Fermi energy, characterized by spin,valley, and orbital quantum numbers. We incorporate in our study two effectsnot previously considered: (i) the nonperturbative effect of trigonal warpingin the single-particle Hamiltonian, and (ii) short-range SU(4)symmetry-breaking interactions that distinguish the energetics of the orbitals.We find within this model a rich set of phases, including ferromagnetic,layer-polarized, canted antiferromagnetic, Kekul\'e, a "spin-valley entangled"state, and a "broken U(1) $\times$ U(1)" phase. This last state involvesindependent spontaneous symmetry breaking in the layer and valley degrees offreedom, and has not been previously identified. We present phase diagrams as afunction of interlayer bias $D$ and perpendicular magnetic field $B_{\perp}$for various interaction and Zeeman couplings, and discuss which are likely tobe relevant to BLG in recent measurements. Experimental properties of thevarious phases and transitions among them are also discussed.

Spin-Valley Coherent Phases of the $\nu=0$ Quantum Hall State in Bilayer Graphene