Collective excitations of a BCS superfluid in the presence of two sublattices

TL;DR

This paper analyzes collective excitations in a lattice-based BCS superfluid, revealing how Goldstone and Leggett modes behave and highlighting the geometric contribution to the Goldstone mode velocity.

Contribution

It introduces a comprehensive analysis of collective modes in a two-sublattice BCS superfluid, including the geometric effects on Goldstone mode velocity.

Findings

Goldstone mode is gapless and propagating at zero momentum.

Leggett mode becomes undamped with strong interactions.

Goldstone mode velocity has a geometric contribution from the quantum metric.

Abstract

We consider a generic Hamiltonian that is suitable for describing a uniform BCS superfluid on a lattice with a two-point basis, and study its collective excitations at zero temperature. For this purpose, we first derive an effective-Gaussian action for the pairing fluctuations, and then extract the low-energy dispersion relations for the in-phase Goldstone and out-of-phase Leggett modes along with the corresponding amplitude (i.e., the so-called Higgs) ones. We find that while the Goldstone mode is gapless at zero momentum and propagating in general, the Leggett mode becomes undamped only with sufficiently strong interactions. Furthermore, we show that, in addition to the conventional contribution that is controlled by the energy of the Bloch bands, the velocity of the Goldstone mode has a geometric contribution that is governed by the quantum metric tensor of the Bloch states. Our results suggest that the latter contribution dominates the velocity when the former one becomes negligible for a narrow- or a flat-band.