General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas
D. T. Son, M. Wingate
TL;DR
This work develops a nonrelativistic effective field theory for unitary Fermi gases by exploiting general coordinate and conformal invariance. By coupling nonrelativistic particles to external gauge fields and a curved spatial metric, the authors derive a general coordinate invariant framework and show how conservation laws follow from these symmetries. At leading order, the superfluid EFT reduces to Thomas–Fermi theory and hydrodynamics, with a single parameter $\xi$ linking the equation of state to the EFT coefficient $c_0$. At next-to-leading order, two additional dimensionless constants $c_1$ and $c_2$ appear, governing derivative corrections to observables such as phonon dispersion and density response; loop corrections start at $O(p^4)$, ensuring a controlled expansion. For the unitary gas, conformal invariance is exact, constraining the form of the EFT and enabling predictions for phonon properties, response functions, and trapped gas energetics in terms of a small set of universal constants $\xi$, $c_0$, $c_1$, and $c_2$.
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.