$ϵ$ Expansion for a Fermi Gas at Infinite Scattering Length

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 $ϵ=4-d$, where $d$ is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\mu /{\epsilon }_F=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\ln ϵ-0.0246{ϵ}^{5/2}+\cdots$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\Delta /\mu =2{ϵ}^{-1}-0.691+\cdots$. The minimum of the fermion dispersion curve is located at $|\mathit{p}|=(2m{\epsilon }_0{)}^{1/2}$, where ${\epsilon }_0/\mu =2+O(ϵ)$. Extrapolation to $d=3$ gives results consistent with Monte Carlo simulations.

Figure 1

Feynman rules. The two vertices on the last column come from ${\mathcal{L}}_2$, while the rest from ${\mathcal{L}}_1$. Solid (dotted) lines represent the fermion (boson) propagator $iG$ $(iD)$.

Figure 2

Restoration of naive $ϵ$ counting for the boson self-energy and the cancellation of tadpole diagrams. The fermion loop in (c) goes around clockwise and counterclockwise.

Figure 3

Vacuum diagrams for the effective potential up to next-to-leading order in $ϵ$. The second diagram is $O(1)$ instead of naive $O(ϵ)$ because of the $1/ϵ$ singularity.

Figure 4

One-loop diagrams for the fermion self-energy of order $O(ϵ)$.