Correlation lengths of flat-band superconductivity from quantum geometry

TL;DR

This paper addresses how characteristic superconducting length scales behave in flat-band systems where kinetic energy is quenched. Using a two-band all-flat-band Hubbard model with uniform pairing, it solves the exact two-body bound-state problem and analyzes zero-temperature Cooper pairing within mean-field theory, complemented by Gaussian-fluctuation results for the coherence length, with Creutz ladder and χ lattice as representative benchmarks. The main finding is that the two-body bound-state size and the Cooper-pair size are finite and governed by the quantum metric of the flat bands, while the zero-temperature coherence length diverges in the dilute limit and near insulating phases, revealing that pair size and coherence length are distinct geometric versus collective scales in flat-band superconductivity. This work clarifies the geometric origin of pairing length scales, reconciles conflicting claims in the literature, and highlights the central role of band geometry in shaping superconductivity when kinetic energy is suppressed.

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 $χ$ 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.

Figures (4)

• Figure 1: Illustration of (a) the Creutz ladder and (b) the $\chi$ lattice. Blue circles (A) and red squares (B) denote the two sublattices, and the lattice spacing is denoted by $a$. In the $\chi$ lattice, each lattice site hosts two orbitals (A and B), and long-range hopping processes connect A and B orbitals on different lattice sites, as indicated by the arrows.
• Figure 2: The order parameter $\Delta_0$ is shown in (a) for the Creutz ladder and in (c) for the $\chi$ lattice as functions of the particle filling $F$ and the interaction strength $U$. The corresponding line cuts are shown in (b) for the Creutz ladder and in (d) for the $\chi$ lattice, where $\Delta_0$ is plotted with solid lines and markers, and the chemical potential $\mu$ is shown with dashed lines. At half filling ($F = 1$), $\Delta_0$ vanishes and $\mu$ is pinned within a finite interval, signaling an insulating state, for interaction strengths below a critical threshold $U < U_c$. The red dots mark the location of $U_c = 2\epsilon$.
• Figure 3: The average Cooper-pair size $\xi_\mathrm{Cp}$ is shown in (a) for the Creutz ladder and in (c) for the $\chi$ lattice as functions of the particle filling $F$ and the interaction strength $U$. In panels (b) and (d), we compare $\xi_\mathrm{Cp}$ with the size of the lowest-bound two-body state $\xi_{2b}$ for the Creutz ladder and the $\chi$ lattice, respectively. Note that $\xi_\mathrm{Cp}$ reduces to $\xi_{2b}$ in the dilute limit for all $U$. Parameter regimes in which $\Delta_0 = 0$ are omitted.
• Figure 4: The zero-temperature coherence length $\xi_0$ is shown in (a) for the Creutz ladder and in (c) for the $\chi$ lattice as functions of the particle filling $F$ and the interaction strength $U$. Regions where $\xi_0^2 < 0$ are shown in white, indicating that the coherence length is ill defined. In panels (b) and (d), we show representative line cuts for the Creutz ladder and the $\chi$ lattice, respectively. The coherence length $\xi_0$ diverges both in the dilute limit and in the vicinity of the insulating regime at half filling.