ϵExpansion for a Fermi Gas at Infinite Scattering Length

Abstract

We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, where $d$ is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}$. The minimum of the fermion dispersion curve is located at $|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}$, where ${\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)$. Extrapolation to $d=3$ gives results consistent with Monte Carlo simulations.