Thermodynamics of the BCS-BEC crossover

We present a self-consistent theory for the thermodynamics of the BCS-BEC crossover in the normal and superfluid phase which is both conserving and gapless. It is based on the variational many-body formalism developed by Luttinger and Ward and by DeDominicis and Martin. Truncating the exact functional for the entropy to that obtained within a ladder approximation, the resulting self-consistent integral equations for the normal and anomalous Green functions are solved numerically for arbitrary coupling. The critical temperature, the equation of state, and the entropy are determined as a function of the dimensionless parameter $1∕k_{F}a$, which controls the crossover from the BCS regime of extended pairs to the BEC regime of tightly bound molecules. The tightly bound pairs turn out to be described by a Popov-type approximation for a dilute, repulsive Bose gas. Even though our approximation does not capture the critical behavior near the continuous superfluid transition, our results provide a consistent picture for the complete crossover thermodynamics which compares well with recent numerical and field-theoretic approaches at the unitarity point.

Figure 1

The functional $\Phi \left[G\right]$ in self-consistent ladder approximation. The propagator lines are dressed lines identified by the matrix Green function $G$.

Figure 2

The functional ${\mathcal{K}}^{\left(2\right)}[G,\Gamma ]$ is the sum of all two-line irreducible diagrams. The propagator lines and the vertices (solid circles) are dressed and identified with $G$ and $\Gamma$, respectively.

Figure 3

(Color online) Modifications of the dimensionless interaction parameter: the red solid curve shows $\delta v_{\mathrm{mod}}$ and the blue dashed curve depicts $\delta {\tilde{v}}_{\mathrm{mod}}$ as a function of $v$ for $T=0$ and $\mid \Delta \mid =\mid {\Delta }_0\mid$.

Figure 4

(Color online) ${\theta }_c^{\left(lower\right)}$ (red dashed line) and ${\theta }_c^{\left(upper\right)}$ (solid black line, identified as $T_c$) compared with the Shohno result (blue dot-dashed line) with $a_{dd}^{\left(B\right)}=2a$ and the exact (QMC) result (light-blue squares) with $\mathrm{\Delta }{T}_c∕T_{BEC}=c{n}_B^{1∕3}{a}_{dd}$and $c=1.31$ and $a_{dd}=0.60a$. The yellow dashed line and green triangles show the BCS result without and with Gorkov and Melik-Barkhudarov corrections.

Figure 5

(Color online) $S\left(T\right)$ at various interaction strengths $v$.

Figure 6

(Color) Entropy as a function of $\theta$ and $v$ obtained using Eqs. (2.50, 2.63).

Figure 7

(Color) Pressure as a function of $\theta$ and $v$ obtained using Eqs. (2.48, 2.58).

Figure 8

(Color online) 3D view of the order parameter.

Figure 9

(Color online) Isothermal sound speed $m{c}^2=\partial p∕\partial n$ as a function of $v=1∕k_{F}a$ for $T=0$. The different curves are explained in the main text.

Figure 10

Internal energy at unitarity as a function of temperature calculated using Eqs. (2.49, 2.62). The dashed curve is obtained from the calculated pressure using the scaling formula $U=\frac{3}{2}pV$ valid at unitarity.

Figure 11

The single-particle chemical potential at unitarity as a function of temperature obtained from the number conservation constraint (2.26).

Figure 12

Entropy at unitarity as a function of temperature.

Figure 13

Temperature reduction on performing an isentropic sweep across $v=0$ from $v=2$ to $v=-2$.

Figure 14

Entropy at $T_c$ as a function of $v=1∕k_{F}a$. Numerical result (solid) line obtained with Eqs. (2.50, 2.63) compared with the limiting results: the BCS mean-field result (triangles) from Eq. (2.50) and (dashed line) from Eq. (3.13) and the Shohno-Popov result (dot-dashed line) from Eq. (2.84).