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Scenario for Fractional Quantum Hall Effect in Bulk Isotropic Materials

We investigate the possibility of a strongly correlated Fractional Quantum Hall (FQH) state in bulk three dimensional isotropic (not layered) materials. We find that a FQH state can exist at low densities only if it is accompanied by a staging transition in which the electrons re-organize themselves in layers, perpendicular to the magnetic field, at distances of order the magnetic length apart. The Hartree energy associated to the staging transition is off-set by the correlation Fock energy of the 3D FQH state. We obtain the phase diagram of bulk electrons in a magnetic field subject to Coulomb interactions as a function of carrier density and lattice constant. At very low densities, the 3D FQH state exhibits a transition to a 3D Wigner crystal state stabilized by phonon correlations.