Vortex States and Coherence Lengths in Flat-Band Superconductors

TL;DR

This paper explores how vortex states in flat-band superconductors reveal quantum geometric effects, showing that vortex size and bound states are governed by the quantum metric, providing universal probes of quantum geometry.

Contribution

It introduces vortex states as direct probes of quantum geometry in flat-band superconductors and analyzes how quantum metric influences vortex properties.

Findings

Vortex bound states are sharply localized near the core.

Vortex size at zero temperature is atomic-scale and quantum metric-dependent.

Vortex size near Tc diverges following Ginzburg-Landau theory.

Abstract

Superconductivity in flat-band systems, governed by quantum metric of Bloch states rather than the BCS framework, exhibits unique phenomena due to the vanishing electron group velocity. Here, we propose the vortex states and vortex size as direct probes to explore the quantum geometry effects in flat-band superconductors. We show that flat-band vortex bound states are sharply localized near the vortex core, and the energy gap between the lowest two bound states is on the order of the bulk superconducting gap. Both the spatial spread and energy scales of bound states are controlled by the flat-band's quantum metric length. Moreover, the vortex size at zero temperature, set by the quantum metric length, is atomic in scale and independent of interaction strength. Near $T_c$, the vortex size corresponds to the Ginzburg-Landau coherence length, diverges as $\xi\sim \sqrt{T_c/(T_c-T)}\xi_0$, where $\xi_0$ depends linearly on the quantum metric length. Thus, the quantum metric serves as the lower bound for vortex state spread and vortex size. We also introduce perturbations to make the flat band dispersive, and distinguish flat-band vortices from BCS-like vortices. Our results establish vortices as universal probes of quantum geometry in flat-band superconductors.