Spin 1/2 Fermions in the Unitary Limit.I

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 δ(k) = π/2 a one-term separable potential is obtained by inverse scattering assuming a momentum cutoff Λ such that δ(k) = 0 for k > Λ. The effective 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 ξ = 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 ξ = 0.24. This result is independent of density and of the cutoff Λ if Λ >∼ 3k f. With m * = 1 there is however a strong dependence on this cutoff. Including hh-ladders estimates give ξ = 0.4 ↔ 0.6, but a reliable result would in this case require a Green's function calculation.