At and near charge neutrality, monolayer graphene in a perpendicular magnetic field is a quantum Hall ferromagnet. In addition to the highly symmetric Coulomb interaction, residual lattice-scale interactions, Zeeman, and sublattice couplings determine the fate of the ground state. Going beyond the simplest model with ultra-short-range residual couplings to more generic couplings, one finds integer phases that show the coexistence of magnetic and lattice order parameters. Here, we show that fractional quantum Hall states in the vicinity of charge neutrality have even richer phase diagrams, with a plethora of phases with simultaneous magnetic and lattice symmetry breaking.

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• Fractional quantum Hall effect
• Phase diagrams
• Ferromagnets
• Graphene