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 $\kappa =\mathbf{(}(e^2/\epsilon \ell )/(\hslash v_F/\ell )\mathbf{)}=(e^2/\epsilon v_F\hslash )$, 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 $\kappa \lesssim 2$. 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 $\kappa$. In the $N=1$ Landau level, fractional quantum Hall states are expected for a smaller range of $\kappa$ 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 $\kappa \sim 0.25–0.75$.

• Received 19 May 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.086401