# The fractional quantum Hall regime in graphene

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In the first part of this work, we describe a theory of the ground states and charge gaps in the fractional quantum Hall states of graphene. The theory relies on knowledge of these properties for filling fractions smaller than one. Then, by the application of two mapping rules, one is able to obtain these properties for fractional quantum Hall states at arbitrary fillings, by conceiving the quantum Hall ferromagnets as vacua on which correlated electrons or correlated holes are added. The predicted charge gaps and phase transitions between different fractional quantum Hall states are in good agreement with recent experiments. In the second part, we investigate the low energy theory for the neutral Landau level of bilayer graphene. We closely analyze the way different terms in the Hamiltonian transform under the action of particle-hole conjugation symmetries, and identify several terms that are relevant in explaining the lack of such symmetry in experiments. Combining an accurate parametrization of the electronic structure of bilayer graphene with a systematic account of the impact of screening we are able to explain the absence of particle-hole symmetry reported in recent experiments. We also study the energetics of fractional quantum Hall states with coherence between n=0 and n=1 cyclotron quantum numbers, and obtain a general formula to map the two-point correlation function from their well-known counterparts made from only n=0 quantum numbers. Bilayer graphene has the potential for realizing these states which have no analogue in other two-dimensional electron systems such as Gallium Arsenide. We apply this formula to describe the properties of the n=0/n=1 coherent Laughlin state which displays nematic correlations.