Anomalous Coherence Length in Superconductors with Quantum Metric

TL;DR

This paper reveals that quantum metric contributes an anomalous, fundamental length scale to the coherence length in superconductors, especially impacting flat-band and moiré graphene systems, beyond traditional BCS theory.

Contribution

It introduces a quantum metric term into the coherence length formula, showing a fundamental length scale in superconductors arising from quantum geometry effects.

Findings

Quantum metric adds an anomalous length scale to coherence length.

In flat bands, coherence length is bounded below by quantum metric.

Quantum geometry significantly influences superconductivity in moiré graphene.

Abstract

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.