Superconductivity beyond band geometry: emergence of pair quantum geometry

Abstract: Quantum geometry shapes the effective mass of Bloch particles through the geometric properties of single-particle states. Here we show that this principle extends to paired states. Starting from a generic multiband Hubbard model, we derive an exact effective-mass theorem for two-body bound states and its many-body counterpart for Cooper pairs near the critical temperature within Gaussian fluctuation theory. In both cases, the inverse effective mass separates into a ``conventional'' band-structure contribution and a new geometric contribution, pair quantum geometry, governed by quantum metrics on the pairing manifold, which becomes nontrivial when pairing is non-uniform across sublattices. In the many-body setting, analytic continuation renders the fluctuation kernel non-Hermitian, producing a biorthogonal pair geometry and a generally complex Cooper-pair effective mass whose imaginary part reflects Landau damping. Exact calculations on one-, two-, and three-dimensional lattice models show that pair quantum geometry can make quantitatively significant contributions to the effective mass. These results establish pair quantum geometry as a fundamental ingredient of superconductivity beyond conventional band geometry.