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arXiv:2608.26099

Exact quantum geometry from sublattice symmetry: Closed-form solution of the quarter-flux Harper-Hofstadter model

Isaac Tesfaye, André Eckardt

cond-mat.quant-gascond-mat.mes-hallquant-phSubmitted Aug 26, 2026• Announced Aug 27, 2026• Revised Sep 22, 2026 (v2)
v1 · Aug 26, 2026v2 · Sep 22, 2026
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Abstract

Quantum geometry has emerged as a guiding principle across atomic and condensed-matter physics, shaping the topological responses of Bloch bands and the stability of the correlated phases they host. Sublattice symmetry, though common among bipartite lattice models, has not yet been exploited to obtain closed-form quantum geometry in multiband systems. For this purpose, we derive a general expression for the QGT of sublattice-symmetric systems in terms of contributions from the individual sublattice sectors, and show that this symmetry renders the Bloch Hamiltonian of a paradigmatic four-band model, the quarter-flux Harper-Hofstadter model, anti-block-diagonal, analytically yielding the spectrum, eigenstates, and full quantum geometric tensor (QGT), including the Berry curvature and quantum metric, for all four bands. The model, describing charged particles on a two-dimensional square lattice subjected to a uniform magnetic field, has recently been realized experimentally with ultracold atoms, photons, and superconducting circuits. Finally, we evaluate fractional-Chern-insulator stability criteria analytically and quantify the lowest band of the quarter-flux Harper-Hofstadter model to be a nearly ideal Chern band. Our approach opens a route for studying also the quantum geometry of other sublattice-symmetric multiband systems.

Comments
10 + 1 + 18 pages, 2 + 0 + 4 figures. Comments are welcome

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Quantum geometry has emerged as a guiding principle across atomic and condensed-matter physics, shaping the topological responses of Bloch bands and the stability of the correlated phases they host. Sublattice symmetry, though common among bipartite lattice models, has not yet been exploited to obtain closed-form quantum geometry in multiband systems. For this purpose, we derive a general expression for the QGT of sublattice-symmetric systems in terms of contributions from the individual sublattice sectors, and show that this symmetry renders the Bloch Hamiltonian of a paradigmatic four-band model, the quarter-flux Harper-Hofstadter model, anti-block-diagonal, analytically yielding the spectrum, eigenstates, and full quantum geometric tensor (QGT), including the Berry curvature and quantum metric, for all four bands. The model, describing charged particles on a two-dimensional square lattice subjected to a uniform magnetic field, has recently been realized experimentally with ultracold atoms, photons, and superconducting circuits. Finally, we evaluate fractional-Chern-insulator stability criteria analytically and quantify the lowest band of the quarter-flux Harper-Hofstadter model to be a nearly ideal Chern band. Our approach opens a route for studying also the quantum geometry of other sublattice-symmetric multiband systems.
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We study how the stability of the fractional quantum Hall effect is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria play a dominant role in predicting the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.
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Understanding the relationship between quantum geometry and topological invariants is a central problem in the study of topological states. In this work, we establish the relationship between the quantum metric and the Euler curvature in two-dimensional systems with space-time inversion IST{I}_{ST}IST​ symmetry satisfying IST2=+1{I}_{ST}^{2}=+1IST2​=+1. As IST{I}_{ST}IST​ symmetry imposes the reality of the wave function with vanishing Berry curvature, the well-known inequality between the quantum metric and the Berry curvature is not meaningful in this class of systems. We find that the non-Abelian quantum geometric tensor of two real bands exhibits an intriguing inequality between the off-diagonal Berry curvature and the quantum metric, which in turn gives the inequality between the quantum volume and the Euler invariant. Moreover, we show that the saturation condition of the inequality is deeply related to the ideal condition for Euler bands, which provides a criterion for the stability of fractional topological phases in interacting Euler bands. Our findings demonstrate the potential of the quantum geometry as a powerful tool for characterizing symmetry-protected topological states and their interaction effect.
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