Effects of Landau Level Mixing on the Fractional Quantum Hall Effect in Monolayer Graphene

We report results of exact diagonalization studies of the spin- and valley-polarized fractional quantum Hall effect in the $N=0$ and $N=1$ Landau levels in graphene. We use an effective model that incorporates Landau level mixing to lowest order in the parameter $\ensuremath{\kappa}=\mathbf{(}({e}^{2}/\ensuremath{\epsilon}\ensuremath{\ell})/(\ensuremath{\hbar}{v}_{F}/\ensuremath{\ell})\mathbf{)}=({e}^{2}/\ensuremath{\epsilon}{v}_{F}\ensuremath{\hbar})$, which is magnetic field independent and can only be varied through the choice of substrate. We find Landau level mixing effects are negligible in the $N=0$ Landau level for $\ensuremath{\kappa}\ensuremath{\lesssim}2$. In fact, the lowest Landau level projected Coulomb Hamiltonian is a better approximation to the real Hamiltonian for graphene than it is for semiconductor based quantum wells. Consequently, the principal fractional quantum Hall states are expected in the $N=0$ Landau level over this range of $\ensuremath{\kappa}$. In the $N=1$ Landau level, fractional quantum Hall states are expected for a smaller range of $\ensuremath{\kappa}$ and Landau level mixing strongly breaks particle-hole symmetry, producing qualitatively different results compared to the $N=0$ Landau level. At half filling of the $N=1$ Landau level, we predict the anti-Pfaffian state will occur for $\ensuremath{\kappa}\ensuremath{\sim}0.25\char21{}0.75$.