Unitary Fermi gas at finite temperature in the $ϵ$ expansion

Thermodynamics of the unitary Fermi gas at finite temperature is investigated from the perspective of the expansion over $ϵ=4-d$ with $d$ being the dimensionality of space. We show that the thermodynamics is dominated by bosonic excitations in the low temperature region $T\ll T_c$. Analytic formulas for the thermodynamic functions as functions of the temperature are derived to the lowest order in $ϵ$ in this region. In the high temperature region where $T\sim T_c$, bosonic and fermionic quasiparticles are excited. We determine the critical temperature $T_c$ of the superfluid phase transition and the thermodynamic functions around $T_c$ to the leading and next-to-leading orders in $ϵ$.

Figure 1

Restoration of naive $ϵ$ counting for the boson self-energy. The fermion loop in (c) goes around clockwise and counterclockwise. Solid (dotted) lines represent the fermion (boson) propagator $-G$ $(-D)$, while the cross in (c) represents the $\mu$ insertion to the fermion propagator.

Figure 2

Boson’s self-energy diagrams contributing to the order $O\left(ϵ\right)$. The vertex ${\Pi }_0$ from ${\mathcal{L}}_2$ needs to be added to the second diagram. The last diagram gives the off-diagonal part of the self-energy. Other elements are given by ${\Pi }_{22}\left(p\right)={\Pi }_{11}(-p)$ and ${\Pi }_{21}\left(p\right)={\Pi }_{12}(p{)}^{*}$.

Figure 3

One-loop diagram of boson contributing to the effective potential at finite temperature. The dotted double line represents the resummed boson propagator $\mathcal{D}$ in Eq. (23).

Figure 4

The behavior of the condensate $\varphi$ as a function of the temperature $T$ to the lowest order in $ϵ$. ${\varphi }_0={\mu }_B∕ϵ$ is the value of the condensate at $T=0$ and the critical temperature is located at $T_c∕{\varphi }_0=1∕\left(2\ln 2\right)\approx 0.721348$.

Figure 5

Three types of diagrams contributing to the pressure up to the next-to-leading order in $ϵ$. Each $\mu$ $\left({\mu }_{\mathrm{B}}\right)$ insertion to the fermion (boson) line reduces the power of $ϵ$ by one.

Figure 6

A type of diagrams resummed to resolve the infrared singularity in Eq. (67).

Figure 7

(Color online) The critical temperature $T_c$ at unitarity as a function of the spatial dimension $d$. The upper solid line is from the expansion around $d=4$ in Eq. (75), while the lower solid curve is from the expansion around $d=2$ in Eq. (78). The middle two curves show the different Borel-Padé approximations connecting the two expansions. The symbols at $d=3$ indicate the results from the Monte Carlo simulations; $T_c∕{\epsilon }_F=0.23\left(2\right)$ [33] (circle), $T_{\mathrm{c}}∕{\epsilon }_F<0.14$ [34] (down arrow), $T_c∕{\epsilon }_F=0.152\left(7\right)$ [35] (diamond), and $T_c∕{\epsilon }_F\approx 0.25$ [36] (square).