Universal properties of Fermi gases in arbitrary dimensions

TL;DR

This paper derives universal relations for spin-1/2 Fermi gases in any dimension, revealing how confinement and dimensionality influence scattering properties and contact relations without solving complex scattering problems.

Contribution

It generalizes Tan's relations and scattering descriptions to arbitrary dimensions, including non-integer, and uncovers universal behaviors of Fermi gases under confinement.

Findings

Universal relations hold in arbitrary dimensions.

Confinement-induced resonances occur in all dimensions except D=2.

Reduced-dimensional contacts relate to 3D contact via geometric factors.

Abstract

We consider spin-1/2 Fermi gases in arbitrary, integer or non-integer spatial dimensions, interacting via a Dirac delta potential. We first generalize the method of Tan's distributions and implement short-range boundary conditions to arbitrary dimension and we obtain a set of universal relations for the Fermi gas. Three-dimensional scattering under very general conditions of transversal confinement is described by an effectively reduced-dimensional scattering length, which we show depends on the three-dimensional scattering length in a universal way. Our formula for non-integer dimensions interpolates between the known results in integer dimensions 1, 2 and 3. Without any need to solve the associated multichannel scattering problem, we find that confinement-induced resonances occur in all dimensions different from D=2, while reduced-dimensional contacts, related to the tails of the momentum distributions, are connected to the three-dimensional contact by a correction factor of purely geometric origin.