Anomalous coherence length in superconductors with quantum metric

The coherence length ξ is the fundamental length scale of superconductors which governs the sizes of Cooper pairs, vortices, Andreev bound states and more. In existing microscopic theories of superconductivity, it is expected that as the strength of attractive interaction increases, ξ decreases as the electrons are bound together more strongly. In BCS theory, the coherence length is ξ_BCS =v_F/Δ, where v_F is the Fermi velocity and Δ is the pairing gap. It is clear that increasing Δ will shorten ξ_BCS. However, the situation is puzzling for superconductors with completely flat bands in which v_F goes to zero and ξ_BCS is expected to be zero. We show that the quantum metric, which is the real part of the quantum geometric tensor, gives rise to an anomalous contribution to the coherence length. Specifically, ξ² = ξ_BCS²+l_qm² for a superconductor where l_qm is the quantum metric contribution. In the flat band limit, ξ does not vanish but bound by l_qm. Incredibly, for the nontrivial flat bands with Chern number C, ξ has a topological bound by C. Physically, the Cooper pair size of a superconductor cannot be squeezed down to a size smaller than l_qm which is a fundamental length scale determined by the quantum geometry of the bands. Lastly, we compute the quantum metric contributions for the family of superconducting moiré graphene materials, demonstrating the significant role played by quantum metric effects in these narrow-band superconductors.