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Fractional and integer quantum Hall effects in the zeroth Landau level in graphene

Experiments on the fractional quantized Hall effect in the zeroth Landau level of graphene have revealed some striking differences between filling factors in the ranges 0<|ν|<1 and 1<|ν|<2. We argue that these differences can be largely understood as a consequence of the effects of terms in the Hamiltonian which break SU(2) valley symmetry, which we find to be important for |ν|<1 but negligible for |ν| >1. The effective absence of valley anisotropy for |ν|>1 means that states with odd numerator, e.g. |ν|=5/3 or 7/5 can accommodate charged excitations in the form of large valley skyrmions, which have a low energy cost, and may be easily induced by coupling to impurities. The absence of observed quantum Hall states at these fractions is likely due to the effects of valley skyrmions. For |ν|<1, the anisotropy terms favor phases in which electrons occupy states with opposite spins, similar to the antiferromagnetic state previously hypothesized to be the ground state at ν=0. The anisotropy and Zeeman energies suppress large-area skyrmions, so that quantized Hall states can be observable at a set of fractions similar to those in GaAs two-dimensional electron systems. In a perpendicular magnetic field B, the competition between the Coulomb energy, which varies as B^{1/2}, and the Zeeman energy, which varies as B, can explain the observation of apparent phase transitions as a function of B for fixed ν, as transitions between states with different degrees of spin polarization. In addition to an analysis of various fractional states, and an examination of the effects of disorder on valley skyrmions, we present new experimental data for the energy gaps at integer fillings ν=0 and ν= -1, as a function of magnetic field, and we examine the possibility that valley-skyrmions may account for the smaller energy gaps observed at ν= -1.