Finite-Temperature Equation of State of Polarized Fermions at Unitarity

We study in a nonperturbative fashion the thermodynamics of a unitary Fermi gas over a wide range of temperatures and spin polarizations. To this end, we use the complex Langevin method, a first principles approach for strongly coupled systems. Specifically, we show results for the density equation of state, the magnetization, and the magnetic susceptibility. At zero polarization, our results agree well with state-of-the-art results for the density equation of state and with experimental data. At finite polarization and low fugacity, our results are in excellent agreement with the third-order virial expansion. In the fully quantum mechanical regime close to the balanced limit, the critical temperature for superfluidity appears to depend only weakly on the spin polarization.

Figure 1

(Top) Density of the balanced UFG obtained by CL (blue squares), in units of the noninteracting unpolarized density $n_0$ as a function of the dimensionless average chemical potential $\beta \mu$. Also shown, third-order virial expansion (dashed line), experimental results of Refs. [83, 87] (red circles), and theoretical results obtained by bold diagrammatic Monte Carlo (BDMC) calculations [87] (dark diamonds) and determinantal hybrid Monte Carlo (DHMC) calculations [82] (light diamonds). (Bottom) Compressibility $\kappa$ as derived from the density EOS (see Supplemental Material [93]) in units of its noninteracting ground-state value ${\kappa }_0$, as a function of the pressure $P$ normalized by the noninteracting ground-state pressure $P_0$ (blue squares), compared to experimental values [83] (red circles) and third-order virial expansion (dashed line). Statistical uncertainties for the CL results are on the order of the symbol sizes. Shaded areas indicate the superfluid phase.

Figure 2

(Left) Density of the UFG in units of the noninteracting density from bottom to top: $\beta h=0$ (circles), 0.4 (octagons), 0.8 (hexagons), 1.2 (pentagons), 1.6 (squares), 2.0 (triangles), compared to the third-order virial expansion (dashed lines). Colors encode fixed values of $\beta \mu$ shown in all panels. (Center) Magnetization in units of the interacting density for the balanced system as a function of $\beta h$ for several values of $\beta \mu$. For $\beta \mu \le -1.0$, third-order virial expansion is shown with dashed lines. (Right) Dimensionless magnetic susceptibility ${\overline{\chi }}_M$ as a function of $\beta h$ (symbols) compared to the corresponding susceptibility of the free Fermi gas ${\overline{\chi }}_M^0$ (dotted lines) at equal chemical potential and asymmetry (color and shape coding as in other panels). (Inset) Ratio ${\overline{\chi }}_M/{\overline{\chi }}_M^0$ as a function of $\beta \mu$ at $\beta h=0.4$.