BEC-BCS crossover in the ϵ expansion

Abstract

The $ϵ$ expansion (expansion around four spatial dimensions) developed by Nishida and Son for a cold Fermi gas with infinite scattering length is extended to finite scattering length to study the Bose-Einstein condensate (BEC) to BCS crossover. A resummation of higher-order logarithms and a suitable extension of fermion coupling in $d$ dimensions are developed in order to apply the theory in the BCS regime. The ratio between the chemical potential and the Fermi energy, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}$, is computed to next-to-leading order in the $ϵ$ expansion as a function of $\ensuremath{\eta}=1∕(a{k}_{F})$, where $a$ is the scattering length and ${k}_{F}$ is the Fermi momentum in a noninteracting system. Near the unitarity limit $\ensuremath{\mid}\ensuremath{\eta}\ensuremath{\mid}\ensuremath{\rightarrow}0$, we found $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.475\ensuremath{-}0.707\ensuremath{\eta}\ensuremath{-}0.5{\ensuremath{\eta}}^{2}$. Near the BEC limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.062∕\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{2}$, while near the BCS limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=1+0.707∕\ensuremath{\eta}$. Overall good agreement with quantum Monte Carlo results is found.