Pair size and quantum geometry in a multiband Hubbard model

TL;DR

This paper investigates how pair size in a multiband Hubbard model is influenced by quantum geometry, revealing geometric contributions to pair size and demonstrating differences between dispersive and flat-band regimes.

Contribution

It introduces a method to include quantum-metric tensor effects in pair size calculations and applies it to a pyrochlore-Hubbard model, showing geometric effects are significant.

Findings

Pair size diverges in weakly interacting dispersive bands.

Pair size remains finite and small in flat-band regimes.

Quantum geometry significantly influences pair size calculations.

Abstract

We study the size of two-body bound states and Cooper pairs within a multiband Hubbard model that features time-reversal symmetry and uniform pairing on a generic lattice. Our analysis involves (i) an exact calculation of the localization tensor to determine the size of lowest-lying two-body bound state in vacuum, and (ii) an evaluation of the analogous tensor to estimate the average size of Cooper pairs within the mean-field BCS-BEC crossover theory at zero temperature. Beyond the conventional intraband contribution that depends on Bloch bands, we show that pair size also has a geometric contribution governed by the quantum-metric tensor of the Bloch states and their band-resolved quantum-metric tensors. As a concrete example, we investigate the pyrochlore-Hubbard model numerically and demonstrate that, while the pair size diverges in the weakly interacting BCS regime of dispersive bands, it remains finite and relatively small in the flat-band regime, even for infinitesimal interaction, perfectly matching the exact two-body result in the dilute limit.