Quantum Geometric Quadrupole of Cooper Pairs

TL;DR

This paper introduces a unified geometric framework for understanding Cooper pair size in superconductors, emphasizing the role of quantum geometry, especially Berry curvature, in flat bands.

Contribution

It develops a general theory based on the Cooper pair quadrupole moment that incorporates Berry curvature and quantum metric, unifying dispersive and flat-band cases.

Findings

Berry curvature significantly influences pair size in flat bands.

The framework predicts pair sizes comparable to experimental coherence lengths.

Berry curvature imposes a geometric lower bound on the Cooper pair size.

Abstract

The size of Cooper pairs defines a fundamental length scale of superconductivity, conventionally set by band dispersion and the superconducting gap. This picture breaks down in flat bands, where quenched dispersion makes quantum geometry essential. Here we develop a general framework based on the Cooper pair quadrupole moment, whose trace gives the pair size. The framework holds for both dispersive and flat-band cases, and provides a unified description of the geometric origin of this length scale. In particular, when time-reversal symmetry is broken, Berry curvature enters through the phase structure of the pair wavefunction and gives an essential contribution absent from previous quantum-metric theories. Together, Berry curvature and quantum metric impose a geometric lower bound on the pair size. Applying this framework to rhombohedral graphene, we find that the Berry-curvature-induced contribution can dominate and yields pair sizes comparable to experimentally inferred coherence lengths. These results identify Berry curvature as a central geometric ingredient controlling the microscopic length scale of superconductivity.