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

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$.