Spin 1/2 Fermions in the Unitary Limit.I

Abstract

This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift $\delta(k)=\pi/2$ a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off $\Lambda$ such that $\delta(k)=0$ for $k>\Lambda$. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass $m^{*}=1$. Using only the zero relative momentum component of this interaction the total energy is $\xi=4/9$ (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to $\xi=0.24.$ This result is independent of density and of the cut-off $\Lambda$ if $\Lambda > \sim 3k_{f}$. With $m^{*}\neq 1$ there is however a strong dependence on this cut-off. Including hh-ladders estimates give $\xi=0.4\leftrightarrow 0.6$, but a reliable result would in this case require a Green's function calculation.