Scattering Hypervolume of Fermions in Two Dimensions

TL;DR

This paper introduces the three-body scattering hypervolume $D_F$ for 2D spin-polarized fermions, deriving its properties, approximate formulas, and effects on energy, pressure, and recombination rates in ultracold gases.

Contribution

It defines and analyzes the scattering hypervolume $D_F$ for 2D fermions, providing asymptotic expansions, an approximate formula, and studying its physical implications.

Findings

Derived asymptotic expansions involving $D_F$

Provided an approximate formula for $D_F$ using Born expansion

Calculated energy and pressure shifts due to $D_F$ in 2D Fermi gases

Abstract

We define the three-body scattering hypervolume $D_F$ for identical spin-polarized fermions in two dimensions, by considering the wave function of three such fermions colliding at zero energy and zero orbital angular momentum. We derive the asymptotic expansions of such a wave function when three fermions are far apart or one pair and the third fermion are far apart, and $D_F$ appears in the coefficients of such expansions. For weak interaction potentials, we derive an approximate formula of $D_F$ by using the Born expansion. We then study the shift of energy of three such fermions in a large periodic area due to $D_F$. This shift is proportional to $D_F$ times the square of the area of the triangle formed by the momenta of the fermions. We also calculate the shifts of energy and of pressure of spin-polarized two-dimensional Fermi gases due to a nonzero $D_F$ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases in two dimensions.