Quantum Geometric Quadrupole of Cooper Pairs in Flat-Band Superconductors

The size of Cooper pairs defines a fundamental length scale of superconductivity, yet conventional BCS theory predicts it to vanish in the flat-band limit. While recent works have established a lower bound set by the quantum metric, the role of the full quantum geometry—including Berry curvature—remains unresolved. Here we develop a general framework for the quadrupole moment of Cooper pairs, whose trace defines the pair size, unifying how band geometry and dispersion govern their spatial extent. We show that Berry curvature and quantum metric together impose a fundamental lower bound on the pair size when time-reversal symmetry is broken. Applied to rhombohedral graphene, the calculated Cooper pair size shows the same order of magnitude as the coherence length reported in the recent experiment on chiral superconductivity [1]. 

[1] Tonghang Han, ... , Long Ju, Signatures of chiral superconductivity in rhombohedral graphene, Nature 643, 654–661 (2025)