An exact solution to the Bertsch problem and the non-universality of the Unitary Fermi Gas

TL;DR

This paper provides an exact analytical solution to the Bertsch problem, demonstrating the non-universality of the Unitary Fermi Gas by showing that the Bertsch parameter varies with the effective range.

Contribution

The authors derive an exact solution for the Bertsch parameter's dependence on effective range, revealing its non-universality in the ground state of the Unitary Fermi Gas.

Findings

Exact value of $\xi$ for zero effective range: 0.56

Existence of a class of solutions with $\xi$ ranging from 0.56 to -1/3

Demonstration that $\xi$ is not universal and depends on effective range

Abstract

We analyze the universality of the Unitary Fermi Gas in its ground state from a Wilsonian renormalization point of view and compute the effective range dependence of the Bertsch parameter $\xi$ exactly. To this end we construct an effective block-diagonal two-body separable interaction with the Fermi momentum as a cut-off which reduces the calculation to the mean field level. The interaction is separable in momentum space and is determined by Tabakin's inverse scattering formula. For a vanishing effective range we get $\xi = \frac{176}{9 \pi }-\frac{17}{3} = 0.56$. By using phase-equivalent similarity transformations we can show that there is a class of exact solutions with any value in the range $ 0.56 \ge \xi \ge -1/3$.