Quantum anomalous Hall state in bilayer graphene

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 renormalization group (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.