Pressure, Compressibility, and Contact of the Two-Dimensional Attractive Fermi Gas

Using ab initio lattice methods, we calculate the finite temperature thermodynamics of homogeneous two-dimensional spin-$1/2$ fermions with attractive short-range interactions. We present results for the density, pressure, compressibility, and quantum anomaly (i.e., Tan’s contact) for a wide range of temperatures (mostly above the superfluid phase, including the pseudogap regime) and coupling strengths, focusing on the unpolarized case. Within our statistical and systematic uncertainties, our prediction for the density equation of state differs quantitatively from the prediction by Luttinger-Ward theory in the strongly coupled region of parameter space, but otherwise agrees well with it. We also compare our calculations with the second- and third-order virial expansion, with which they are in excellent agreement in the low-fugacity regime.

Figure 1

Density equation of state, in units of the noninteracting density $n_0$ of spin-$1/2$ fermions in 2D, for coupling strengths $\beta {ϵ}_B=0.1$, 0.5, 1, 2, 3 (from bottom to top), as a function of $\beta \mu$. The error bars reflect the statistical uncertainty. The solid colored lines show the Luttinger-Ward result of Ref. [31]. The long- and short-dashed lines show the second- and third-order virial expansion results, respectively.

Figure 2

Pressure, in units of the noninteracting pressure $P_0$, of spin-$1/2$ fermions in 2D for coupling strengths $\beta {ϵ}_B=0.1$, 0.5, 1, 2, 3 (from bottom to top), as a function of $\beta \mu$. The error bars reflect the statistical uncertainty. The long-and short-dashed lines show the second- and third-order virial expansion results, respectively.

Figure 3

Compressibility, in units of the noninteracting compressibility ${\kappa }_0$, of spin-$1/2$ fermions in 2D for coupling strengths $\beta {ϵ}_B=0.1$, 0.5, 1, 2, 3 (from top to bottom), as a function of $\beta \mu$. The error bars reflect the difference between a smooth interpolation and the raw LMC data. The long- and short-dashed lines show the second-and third-order virial expansion results, respectively.

Figure 4

Contact, in units of $1/(N{k}_F^2)$, for spin-$1/2$ fermions in 2D for coupling strengths $\beta {ϵ}_B=0.1$, 0.5, 1, 2, 3 (from bottom to top), as a function of $T/T_F$. The error bars reflect the statistical uncertainty. Solid lines of gray across colored contact results indicate lines of constant $\ln (k_F{a}_{2\mathrm{D}})=1.60,1.21,0.85,0.53,0.24$ from bottom to top. Corresponding experimental data at $T/T_F=0.27$ from Ref. [18] given by solid circles (displaced about light gray line for visibility of error bars); corresponding Luttinger-Ward results at $T/T_F=0.27$ from Ref. [31] are shown as black stars; and corresponding QMC calculations from Ref. [28] at $T/T_F=0$ given by solid squares and dotted gray lines.