Correlation lengths of flat-band superconductivity from quantum geometry

TL;DR

This paper explores how quantum geometry influences superconducting length scales in flat-band systems, revealing that pair size and coherence length behave differently, with implications for understanding superconductivity when kinetic energy is suppressed.

Contribution

It demonstrates that in flat-band superconductors, pair size and coherence length are distinct and controlled by quantum geometry, with specific behaviors in weak-coupling and insulating regimes.

Findings

Two-body bound-state size remains finite and small in weak coupling

Cooper-pair size is controlled by the quantum metric of flat bands

Coherence length diverges in dilute and insulating regimes

Abstract

Flat-band superconductors provide a regime in which kinetic energy is quenched, so that pairing is governed primarily by interactions and quantum geometry. We investigate characteristic superconducting length scales in all-flat-band systems under the assumptions of time-reversal symmetry and spatially-uniform pairing, focusing on the size of the lowest-lying two-body bound state, the average Cooper-pair size, and the zero-temperature coherence length in two-band Hubbard models. Using the Creutz ladder and the $\chi$ lattice as representative examples, we show that both the two-body bound-state size and the many-body Cooper-pair size remain finite and small in the weak-coupling limit, being controlled by the quantum metric of the flat bands. By contrast, the coherence length exhibits qualitatively distinct behavior, diverging in the dilute limit and in the vicinity of insulating regimes. These results demonstrate that, in flat-band superconductors, the pair size and the coherence length are fundamentally distinct physical quantities and highlight the central role of band geometry in shaping superconducting length scales when kinetic energy is quenched.