  We present a symmetry-based analysis of competition between different gapped
states that have been proposed in bilayer graphene (BLG), which are all
degenerate on a mean field level. We classify the states in terms of a hidden
SU(4) symmetry, and distinguish symmetry protected degeneracies from accidental
degeneracies. One of the states, which spontaneously breaks discrete time
reversal symmetry but no continuous symmetry, is identified as a Quantum
Anomalous Hall (QAH) state, which exhibits quantum Hall effect at zero magnetic
field. We investigate the lifting of the accidental degeneracies by thermal and
zero point fluctuations, taking account of the modes softened under RG. Working
in a 'saddle point plus quadratic fluctuations' approximation, we identify two
types of RG- soft modes which have competing effects. Zero point fluctuations,
dominated by 'transverse' modes which are unique to BLG, favor the QAH state.
Thermal fluctuations, dominated by 'longitudinal' modes, favor a SU(4) symmetry
breaking multiplet of states. We discuss the phenomenology and experimental
signatures of the QAH state in BLG, and also propose a way to induce the QAH
state using weak external magnetic fields.
