The Hubbard-Wigner model: from Aubry-Mather theory to Fractional Quantum Hall Effects

Abstract

The Hubbard-Wigner model describes binary lattice configurations with convex repulsive pair interactions. We give a unified classification of its periodic minimizers, minimizing invariant measures, and Aubry sets through a positive block-count identity, recovering explicit plateau widths and a complete devil's staircase for the minimizing density. This variational structure parallels Aubry-Mather theory and the authors' work on Aubry's completeness conjecture, where the staircase is related to the sliding and pinning mechanisms underlying the conducting and insulating interpretations of particle chains. In the binary model, completeness follows from discrete convexity and an exact sum of plateau widths. We then study how this classical structure persists for interactions arising in the fractional quantum Hall effect on a thin torus. At rational density $p/q$, the diagonal reference has a $q$-dimensional lowest eigenspace, a spectral gap uniform in the number of repeated periods, and an associated bundle of Chern number $p$. For the full $V_1$ operator, we prove persistence of a density-$1/3$ plateau and a positive spectral gap, uniformly in system size at sufficiently small fixed transverse circumference; the lowest-eigenspace bundle has rank three and Chern number one. For the full periodic Coulomb operator, we establish multiplicity, spectral-gap, and Chern-number conclusions at bounded denominators, and quantitative bounds on plateau-endpoint displacement and loss of width under explicit joint conditions on size and geometry.