Fractional Quantum Hall Effects in Graphene and Its Bilayer

Single-layer and bilayer graphene are new classes of two-dimensional electron systems with unconventional band structures and valley degrees of freedom. The ground states and excitations in the integer and fractional quantum Hall regimes are investigated on torus and spherical geometries using the density matrix renormalization group (DMRG) method. At nonzero Landau level indices, the ground states at the effective filling factors 1, 1/3, 2/3, and 2/5 are valley polarized both in single-layer and bilayer graphene. We examine the elementary charge excitations that could couple with the valley degrees of freedom (so-called valley skyrmions). The excitation gaps are calculated and extrapolated to the thermodynamic limit. The largest excitation gap at the effective filling 1/3 is obtained in bilayer graphene, which is a good candidate for the experimental observation of the fractional quantum Hall effect.