Fermi gas near unitarity around four and two spatial dimensions

We construct systematic expansions around four and two spatial dimensions for a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi gas can be understood as a weakly interacting system of fermionic and bosonic degrees of freedom. To the leading and next-to-leading orders in the expansion over $ϵ=4-d$, with $d$ being the dimensionality of space, we calculate the thermodynamic quantities and the fermion quasiparticle spectrum, both at unitarity and around the unitarity point. Then the phase structure of the polarized Fermi gas in the unitary regime is studied in the $ϵ$ expansion. At unitarity the unpolarized superfluid state and a fully polarized normal state are separated by a first order phase transition. However, on the BEC side of the unitarity point, in a certain range of the two-body binding energy, these two phases are separated in the phase diagram by more exotic phases, including gapless superfluid phases with one or two Fermi surfaces and a superfluid phase with a spatially varying condensate. We also show that the unitary Fermi gas near two spatial dimensions is weakly interacting, and calculate the thermodynamic quantities and the fermion quasiparticle spectrum in the expansion over $\overline{ϵ}=d-2$.

Figure 1

Two-fermion scattering in vacuum in the unitarity limit. The $T$ matrix near four spatial dimensions is expressed by the propagation of the boson with the small effective coupling $g$, while it reduces to a contact interaction with the small effective coupling ${\overline{g}}^2$ near two spatial dimensions.

Figure 2

Feynman rules from the Lagrangian density in Eqs. (26, 27, 28). The two vertices in the last column come from ${\mathcal{L}}_2$, while the rest come from ${\mathcal{L}}_1$. Solid (dotted) lines represent the fermion (boson) propagator $iG\left(iD\right)$.

Figure 3

Four apparent exceptions of the naive power counting rule of $ϵ$ [(a), (c), (e), (g)]. The boson self-energy diagram (a) or (c) is combined with the vertex from ${\mathcal{L}}_2$, (b) or (d), to restore the naive $ϵ$ power counting. The tadpole diagram in (e) cancels with other tadpole diagrams at the minimum of the effective potential. The vacuum diagram (g) is the only exception of the naive power counting rule of $ϵ$, which is $O\left(1\right)$ instead of $O\left(ϵ\right)$.

Figure 4

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

Figure 5

Corrections to the fermion self-energy of order $O\left(ϵ\right)$; a $\mu$ insertion and one-loop diagrams.

Figure 6

Illustration of the dispersion relation of the fermion quasiparticle in the unitarity limit from the $ϵ$ expansion. ${\omega }_F\left(\mathbf{p}\right)=\sqrt{{({\epsilon }_{\mathbf{p}}-{\epsilon }_0)}^2+{\Delta }^2}$ in Eq. (72) is plotted as a function of ${\epsilon }_{\mathbf{p}}={\mathbf{p}}^2∕2m$. Values on both axes are extrapolated results to $ϵ=1$ in units of the chemical potential $\mu$.

Figure 7

Schematic phase diagram of the polarized Fermi gas near the unitarity limit from the $ϵ$ expansion. The phase diagram can be divided into four phases, I: the gapped superfluid phase, II: the polarized normal phase, III: the gapless superfluid phase, and IV: the phase with the spatially varying condensate. The inset is the magnification of the region around the splitting point (SP). The phase IV appears in the narrow region represented by the thick line between the phases I and III.

Figure 8

Schematic phase diagram of the polarized Fermi gas near the unitarity limit in the $H\text{−}\mu$ plane for the fixed binding energy ${\epsilon }_b>0$. The phases I, II, and III are the same as in Fig. 7. The horizontal dashed line from right to left tracks the chemical potentials $(\mu ,H)$ in trapped Fermi gases from the center to the peripheral.

Figure 9

Feynman rules for the expansion around two spatial dimensions from the Lagrangian density in Eq. (109). The first line gives propagators, while the second line gives vertices.

Figure 10

Two apparent exceptions of naive power counting rule of $\overline{ϵ}$, [(a), (c)]. The boson self-energy diagram (a) is combined with the vertex from ${\overline{\mathcal{L}}}_2$ (b) to restore the naive $\overline{ϵ}$ power counting. The condition of disappearance of the tadpole diagram (c) gives the gap equation to determine the value of condensate ${\varphi }_0$.

Figure 11

The tadpole diagram which gives the medium-effect correction to the gap equation.

Figure 12

Vacuum diagrams contributing to the pressure up to the next-to-leading order in $\overline{ϵ}$.

Figure 13

One-loop diagrams contributing to the fermion self-energy to order $O\left(\overline{ϵ}\right)$.

Figure 14

An $n\text{th}$ order diagram at $d=2$ which contributes to the pressure as $n\text{!}$ by itself. The countervertex $-i{\overline{\Pi }}_0=i$ for each bubble diagram is understood implicitly.

Figure 15

(Color online) The universal parameter $\xi$ as a function of the spatial dimensions $d$. The upper solid curve is from the expansion around $d=4$ in Eq. (62), while the lower solid line is from the expansion around $d=2$ in Eq. (138). The middle three curves show the different Borel-Padé approximations connecting the two expansions. The diamond indicates the result $\xi \approx 0.42$ from the Monte Carlo simulation [15].