General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas

TL;DR

This paper demonstrates that nonrelativistic particles can be coupled to gauge fields and metrics to exhibit a form of general coordinate invariance, impacting the effective theories of unitary Fermi gases with conformal symmetry.

Contribution

It derives the most general effective Lagrangian for a unitary Fermi gas respecting both nonrelativistic general coordinate and conformal invariance, including next-to-leading order corrections.

Findings

Reproduces Thomas-Fermi and superfluid hydrodynamics at leading order.

Identifies two additional constants at next-to-leading order.

Expresses physical quantities in terms of these constants.

Abstract

We show that the Lagrangian for interacting nonrelativistic particles can be coupled to an external gauge field and metric tensor in a way that exhibits a nonrelativistic version of general coordinate invariance. We explore the consequences of this invariance on the example of the degenerate Fermi gas at infinite scattering length, where conformal invariance also plays an important role. We find the most general effective Lagrangian consistent with both general coordinate and conformal invariance to leading and next-to-leading orders in the momentum expansion. At the leading order the Lagrangian contains one phenomenological constant and reproduces the results of the Thomas-Fermi theory and superfluid hydrodynamics. At the next-to-leading order there are two additional constants. We express various physical quantities through these constants.