Universal relations for a spin-polarized Fermi gas in two dimensions

TL;DR

This paper derives comprehensive universal relations for a two-dimensional spin-polarized Fermi gas with p-wave interactions, highlighting the role of additional contacts and center-of-mass motion effects, and confirms their experimental relevance.

Contribution

It introduces a generalized contact framework for p-wave interactions in 2D Fermi gases, including the effects of center-of-mass motion on high-momentum tails and energy divergences.

Findings

Additional contact related to effective range is necessary for p-wave interactions.

Center-of-mass motions cause extra divergences in kinetic energy.

Universal relations like the virial theorem remain unaffected by pair motions.

Abstract

We derive the full set of universal relations for spin-polarized Fermi gases with $p$-wave interaction in two dimensions, simply using the short-range asymptotic behavior of fermion-pair wave functions. For $p$-wave interactions, an additional contact related to the effective range needs to be introduced, besides the one related to the scattering volume. Since the subleading tail ($k^{-4}$) of the large-momentum distribution cannot fully be captured by the contacts defined by the adiabatic relations, an extra term resulted from the center-of-mass motions of the pairs gives rise to an additional divergence in the kinetic energy of the system, besides those related to the contacts defined. We show in Tan's energy theorem that if only two-body correlations are taken into account, all these divergences are reasonably removed, leading to a finite internal energy of the system. In addition, we find that all the other universal relations, such as the high-frequency behavior of the radio-frequency response, short-range behavior of the pair correlation function, generalized virial theorem, and pressure relation, remain unaffected by the center-of-mass motions of the pairs, and are fully governed by the contacts defined by the adiabatic relations. Our results confirm the feasibility of generalizing the contact theory for higher-partial-wave scatterings, and could readily be confirmed in current experiments with ultracold $^{40}$K and $^{6}$Li atoms.