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After more than three decades the fractional quantum Hall effect still poseschallenges to contemporary physics. Recent experiments point toward a fractalscenario for the Hall resistivity as a function of the magnetic field. Here, weconsider the so-called thin-torus limit of the Hamiltonian describinginteracting electrons in a strong magnetic field, restricted to the lowestLandau level, and we show that it can be mapped onto a one-dimensional latticegas with repulsive interactions, with the magnetic field playing the role of achemical potential. The statistical mechanics of such models leads to interpretthe sequence of Hall plateaux as a fractal phase diagram, whose landscape showsa qualitative agreement with experiments.

Devil's staircase phase diagram of the fractional quantum Hall effect in the thin-torus limit