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More Realistic Hamiltonians for the Fractional Quantum Hall Regime in GaAs and Graphene

We construct an effective Hamiltonian for electrons in the fractional quantum Hall regime for GaAs and graphene that takes into account Landau level mixing (for both GaAs and graphene) and subband mixing (for GaAs, due to the nonzero width of the quantum well). This mixing has the important qualitative effect of breaking particle-hole symmetry as well as renormalizing the strength of the interparticle interactions. Both effects could have important consequences for the prospect that the fractional quantum Hall effect at $ν=5/2$ is described by states that support non-Abelian excitations such as the Moore-Read Pfaffian or anti-Pfaffian states. For GaAs, Landau level and subband mixing break particle-hole symmetry in all Landau levels and subband mixing, due to finite thickness, causes additional short-distance softening of the Coulomb interaction, further renormalizing the Hamiltonian; additionally, the Landau level and subband energy spacings are comparable so it is crucial to consider both effects simultaneously. We find that in graphene, Landau level mixing only breaks particle-hole symmetry outside of the lowest Landau level ($N\neq0$). Landau level mixing is likely to be especially important in graphene since the Landau level mixing parameter is independent of the external magnetic field and is of order one. Our realistic Hamiltonians will serve as starting points for future numerical studies.