| In this thesis, we study the effect of the on-site Hubbard interaction and the long-range Coulomb interaction in monolayer and A-B stacking bilayer graphene by using a non-perturbative projective quantum Monte Carlo method. In the case of monolayer graphene, we find that at weak coupling regime the energy renormalization agrees well with the random phase approximation theory, which depends only on the long-range interaction. In the case of bilayer graphene, we find that the metallic-to-Mott-insulating phase transition occurs only at finite values of interactions, in contrary to the conventional belief that infinitesimally small interactions can induce the phase transition due to instability of quadratic band touching point. For both monolayer and bilayer cases, the short-range Hubbard driven Gross-Neveu physics strongly influences the electronic properties at strong coupling regime. Surprisingly, experimental graphene is located in between these two regimes. Neither side of the theory alone can explain the experimental graphene. |
