‘‘Hall crystal’’ versus Wigner crystal

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Abstract
We consider the possibility of coexistence of the quantized Hall effect (QHE) and a weak charge-density wave. For suitably chosen interactions between the fermions, there is a crystalline ordered state (‘‘Hall crystal’’) exhibiting QHE. Unlike the standard Wigner crystal, the energy of the Hall crystal does not have a cusp minimum when the fermion density is precisely an integer per unit cell of the structure. The density is locked onto a multiple of the magnetic flux density, while the period of the charge-density wave can vary continuously. The normal modes of the Hall crystal show separate transverse and longitudinal modes, with linear dispersion at long wavelengths. We propose a model for which the Hartree approximation is accurate and which exhibits a Wigner-crystal phase, a Hall-crystal phase, and a translationally invariant quantized–Hall-‘‘liquid’’ phase, as the coupling constant is varied. We study the one-particle band structure and its associated Diophantine equations, the Hall conductance, and the normal modes of this model, in its various regimes, exploiting the existence of two topological invariants (Chern numbers) which characterize the system. We also examine the transitions between different phases.