The Colored Hofstadter Butterfly as a Many-Body Quantum Hall Phase Diagram

• URL: http://arxiv.org/abs/2606.26256v1
• Date: Wed, 24 Jun 2026 18:02:28 GMT
• Title: The Colored Hofstadter Butterfly as a Many-Body Quantum Hall Phase Diagram
• Abstract summary: We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems.<n>The integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems. Starting from a spectral gap of a Hofstadter-like one-particle Hamiltonian at arbitrary magnetic flux $b$, we construct an open region in the three-dimensional parameter space $(b,μ,λ)$ of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. The construction combines quasi-adiabatic continuation in the interaction strength with denominator-independent magnetic perturbation estimates, and therefore covers both commensurate and incommensurate fluxes, where no finite magnetic unit cell exists. On connected uniformly gapped regions meeting the non-interacting plane $λ=0$, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying $2πσ^{\mathrm{H}}\in\mathbb{Z}$. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.

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