The 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 1 Landau levels in graphene. We use an effective model that incorporates Landau level mixing to lowest-order in the parameter $\kappa = \frac{e^2/\epsilon\ell}{\hbar v_F/\ell}=\frac{e^2}{\epsilon v_F\hbar}$ 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 $\kappa \lesssim 2$. 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 $\kappa \sim 0.25$-$0.75$.