Density Functional Theory for non-relativistic Fermions in the Unitarity Limit

We derive an energy density functional for non-relativistic spin one-half fermions in the limit of a divergent two-body scattering length. Using an epsilon expansion around $d=4-\epsilon$ spatial dimensions we compute the coefficient of the leading correction beyond the local density approximation (LDA). In the case of $N$ fermionic atoms trapped in a harmonic potential this correction has the form $E=E_{LDA}[ 1 + c_s (3N)^{-2/3} ]$, where $E_{LDA}$ is the total energy in LDA approximation. At next-to-leading order in the $\epsilon$-expansion we find $c_s=1.68$, which is significantly larger than the result for non-interacting fermions, $c_s=0.5$.