Valley symmetry breaking in bilayer graphene: a test to the minimal model

Physical properties reflecting valley asymmetry of Landau levels in a biased bilayer graphene under magnetic field are discussed. Within the $4-$band continuum model with Hartree-corrected self-consistent gap and finite damping factor we predict the appearance of anomalous steps in quantized Hall conductivity due to the degeneracy lifting of Landau levels. Moreover, the valley symmetry breaking effect appears as a non-semiclassical de Haas-van Alphen effect where the reduction of the oscillation period to half cannot be accounted for through quasi-classical quantization of the orbits in reciprocal space, still valley degenerate.