Phase diagram for quantum Hall states in graphene

We investigate integer and half-integer filling states (uniform and unidimensional stripe states, respectively) for graphene using the Hartree-Fock approximation. For fixed filling factor, the ratio between the scales of the Coulomb interaction and Landau level spacing g=(e²/ϵℓ)/(ℏv_F/ℓ), with ℓ as the magnetic length, is a field-independent constant. However, when B decreases, the number of filled negative Landau levels increases, which surprisingly turns out to decrease the amount of Landau level mixing. The resulting states at fixed filling factor ν (for ν not too big) have very little Landau level mixing even at arbitrarily weak magnetic fields. Thus in the density-field phase diagram, many different phases may persist down to the origin, in contrast to the more standard two-dimensional electron gas, in which the origin is surrounded by Wigner crystal states. We demonstrate that the stripe amplitudes scale roughly as B so that the density waves “evaporate” continuously as B→0. Tight-binding calculations give the same scaling for stripe amplitude and demonstrate that the effect is not an artifact of the cut-off procedure used in the continuum calculations.