Anomalous Coherence Length in Superconductors with Quantum Metric

The coherence length $\xi$ 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 $\xi_\mathrm{BCS} = \hbar v_{F}/\Delta$, where $v_{F}$ is the Fermi velocity and $\Delta$ is the pairing gap. It is clear that increasing $\Delta$ will shorten $\xi_\mathrm{BCS}$. 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, $\xi = \sqrt{\xi_\mathrm{BCS}^2 +\ell_{\mathrm{qm}}^{2}}$ for a superconductor where $\ell_{\mathrm{qm}}$ is the quantum metric contribution. In the flat-band limit, $\xi$ does not vanish but is bound below by $\ell_{\mathrm{qm}}$. We demonstrate that under the uniform pairing condition, $\ell_{\mathrm{qm}}$ 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 $\ell_{\mathrm{qm}}$ 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 moir\'{e} graphene materials, demonstrating the significant role played by quantum metric effects in these narrow-band superconductors.