Ground-state energy of the unitary Fermi gas from the $ϵ$ expansion

We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $\left(\xi \right)$ from the $ϵ$ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO $ϵ$ expansions around four and two spatial dimensions with the use of Padé approximants give $\xi \approx 0.360\pm 0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO $ϵ$ expansions $\xi \approx 0.377\pm 0.014$ in spite of the large NNLO corrections.

Figure 1

(Color online) The universal parameter ${\xi }_d$ as a function of spatial dimensions $d$. The upper curve is the extrapolation from the NLO expansion around $d=4$ in Eq. (2), while the lower line is the extrapolation from the NLO expansion around $d=2$ in Eq. (3). The middle three curves show the Padé (left panel) and Borel-Padé [25] (right panel) interpolations of the two NLO expansions. The symbol at $d=3$ indicates the result ${\xi }_3\approx 0.40\left(1\right)$ from the latest Monte Carlo simulations [21, 23].

Figure 2

(Color online) The universal parameter ${\xi }_d$ as a function of spatial dimensions $d$. The lower right curve is the extrapolation from the NNLO expansion around $d=4$ in Eq. (2), while the upper left curve is the extrapolation from the NNLO expansion around $d=2$ in Eq. (3). The middle four curves show the Padé interpolations of the two NNLO expansions. The symbol at $d=3$ indicates the result ${\xi }_3\approx 0.40\left(1\right)$ from the latest Monte Carlo simulations [21, 23].

Figure 3

Power counting rule of $\overline{ϵ}$. The self-energy diagram of $\phi$ field (a) is combined with the vertex from ${\mathcal{L}}_2$ (b) to achieve the simple $\overline{ϵ}$ counting. Solid (dotted) lines represent the fermion (auxiliary field) propagators $iG$ $\left(iD\right)$.

Figure 4

Vacuum diagrams contributing to the pressure to the next-to-next-to-leading order in $\overline{ϵ}$. In the right diagram, the counter vertex in Fig. 3b for each bubble subdiagram is implicitly understood.