Analytic Thermodynamics and Thermometry of Gaudin-Yang Fermi Gases

We study the thermodynamics of a one-dimensional attractive Fermi gas (the Gaudin-Yang model) with spin imbalance. The exact solution has been known from the thermodynamic Bethe ansatz for decades, but it involves an infinite number of coupled nonlinear integral equations whose physics is difficult to extract. Here the solution is analytically reduced to a simple, powerful set of four algebraic equations. The simplified equations become universal and exact in the experimental regime of strong interaction and relatively low temperature. Using the new formulation, we discuss the qualitative features of finite-temperature crossover and make quantitative predictions on the density profiles in traps. We propose a practical two-stage scheme to achieve accurate thermometry for a trapped spin-imbalanced Fermi gas.

Figure 1

Left panel: Comparison of the temperature dependence of the pressure $p$ obtained by numerically solving the full TBA equations (circles) and simplified Eqs. (4, 5, 6, 7) (solid line). The entropy $s(T)$ (green dashed line) is linear in $T$ at low temperatures, agreeing well with $s_{\mathrm{FT}}$ (blue dotted line) as predicted by the low-energy effective field theory. $\mu =-0.95$ and $h=1.1$. The energy and length units are ${\epsilon }_B/2$ and $a_1$, respectively. Right panel: The schematic finite-temperature “phase diagram” in the regime Eq. (3).

Figure 2

The density $n_{\uparrow }$, $n_{\downarrow }$, the magnetization $M=n_{\uparrow }-n_{\downarrow }$, and the compressibility $\kappa =\partial n/\partial \mu$ versus the local chemical potential $\mu$ in a 1D trap. Left panel: low polarization, $h=0.9$. Right panel: high polarization, $h=1.1$. The temperature $T=0.02$ (units are the same as Fig. 1). $\kappa$ develops a peak at the thermal tail of the gas or at the phase boundary. Its peak value depends sensitively on temperature.