# Geometric contribution to the Goldstone mode in spin-orbit coupled Fermi superfluids

## TL;DR

This paper investigates how the quantum metric tensor influences the Goldstone mode in spin-orbit coupled Fermi superfluids, revealing that geometric effects cause nonmonotonic behavior of sound velocity across the BCS-BEC crossover.

## Contribution

It demonstrates that the quantum metric tensor's geometric contribution significantly affects collective excitations, specifically the Goldstone mode, in spin-orbit coupled Fermi superfluids.

## Key findings

- Quantum metric influences superfluid properties related to effective mass.
- Geometric effects cause nonmonotonic sound velocity in BCS-BEC crossover.
- Numerical analysis for Weyl and Rashba couplings supports the geometric impact.

## Abstract

The so-called quantum metric tensor is a band-structure invariant whose measure corresponds to the quantum distance between nearby states in the Hilbert space, characterizing the geometry of the underlying quantum states. In the context of spin-orbit coupled Fermi gases, we recently proposed that the quantum metric has a partial control over all those superfluid properties that depend explicitly on the mass of the superfluid carriers, i.e., the effective-mass tensor of the corresponding (two- or many-body) bound state. Here we scrutinize this finding by analyzing the collective phase and amplitude excitations at zero temperature. In particular to the Goldstone mode, we present extensive numerical calculations for the Weyl and Rashba spin-orbit couplings, revealing that, despite being small, the geometric contribution is solely responsible for the nonmonotonic evolution of the sound velocity in the BCS-BEC crossover.