We numerically study the quantum Hall effect ͑QHE͒ in bilayer graphene based on tight-binding model in the presence of disorder. Two distinct QHE regimes are identified in the full energy band separated by a critical region with nonquantized Hall Effect. The Hall conductivity around the band center ͑Dirac point͒ shows an anomalous quantization proportional to the valley degeneracy, but the = 0 plateau is markedly absent, which is in agreement with experimental observation. In the presence of disorder, the Hall plateaus can be destroyed through the float-up of extended levels toward the band center and higher plateaus disappear first. The central two plateaus around the band center are most robust against disorder scattering, which is separated by a small critical region in between near the Dirac point. The longitudinal conductance around the Dirac point is shown to be nearly a constant in a range of disorder strength, until the last two QHE plateaus completely collapse.

We numerically study the disorder effect on the thermoelectric and thermal transport in bilayer graphene under a strong perpendicular magnetic field. In the unbiased case, we find that the thermoelectric transport has similar properties as in monolayer graphene, i.e., the Nernst signal has a peak at the central Landau level (LL) with a height of the order of kB/e and changes sign near other LLs, while the thermopower has an opposite behavior. We attribute this to the coexistence of particle and hole LLs around the Dirac point. When a finite interlayer bias is applied and a band gap is opened, it is found that the transport properties are consistent with those of a band insulator. We further study the thermal transport from electronic origin and verify the validity of the generalized Weidemann-Franz law.

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In the Van Hove scenario including orthorhombic distortion effect, we develop a pair of coupled gap equations for the mixed (sϩid)-wave order parameter. It is found that a mixed sϩid symmetry state is realized in a certain range of relative strength of the s and d interactions, and there are two second-order transitions between the mixed and the pure symmetry states. Particular attention is paid to the temperature dependence of two components in the mixed order parameter as well as their evolution from a pure s to a pure d symmetry state.