Anomalous coherence length in superconductors with quantum metric

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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 BCS theory, the coherence length is ξBCS = ℏvF/Δ, where vF is the Fermi velocity and Δ is the pairing gap. It is clear that increasing Δ will shorten ξBCS. In this work, 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, ξ=ξBCS2+ℓqm2 for a superconductor where ℓqm is the quantum metric contribution. In the flat-band limit, ξ does not vanish but is bound below by ℓqm. We demonstrate that under the uniform pairing condition, ℓqm is controlled by the quantum metric of minimal trace in the flat-band limit. Physically, the Cooper pair size of a superconductor cannot be squeezed down to a size smaller than ℓqm which is a fundamental length scale determined by the quantum geometry of the wave functions. 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.