Thermodynamics of an Attractive 2D Fermi Gas

Thermodynamic properties of matter are conveniently expressed as functional relations between variables known as equations of state. Here we experimentally determine the compressibility, density, and pressure equations of state for an attractive 2D Fermi gas in the normal phase as a function of temperature and interaction strength. In 2D, interacting gases exhibit qualitatively different features to those found in 3D. This is evident in the normalized density equation of state, which peaks at intermediate densities corresponding to the crossover from classical to quantum behavior.

Figure 1

(a) Average of 10 absorption images of a 2D Fermi gas, with $\beta E_b=0.26$, prepared under the same experimental conditions ($B=880\mathrm{G}$, $N=16,000$, $T\approx 25\mathrm{nk}$) (b) Azimuthally averaged density, $n(V_r)$, obtained from image (a) as a function of the local potential. (c) Dimensionless compressibility $\tilde{\kappa }$ vs dimensionless pressure $\tilde{p}$ of a 2D Fermi gas with $\beta E_b=0.26(2)$. Faint green circles show data extracted from single images and dark green triangles are data binned according to $\tilde{p}$. Grey solid line shows the equation of state for a noninteracting Fermi gas and dashed green line is the predicted $\tilde{\kappa }$ based on the virial expansion to third order.

Figure 2

Normalized density equation of state for a 2D attractive Fermi gas for $\beta E_b=0.47(3)$ (purple circles), $\beta E_b=0.26(2)$ (green triangles), $\beta E_b=0.06(1)$ (blue diamonds), and $\beta E_b=0.005(1)$ (red squares). Dashed lines show the calculated equation of state using the third order virial expansion [49]. Solid blue line and grey stars are the calculated equation of state for $\beta E_b=0.5$ based on Luttinger-Ward (LW) [26] and quantum Monte Carlo (QMC) calculations [27], respectively.

Figure 3

(a) Normalized free energy $F$ and (b) internal energy $U$ as a function of $T/T_F$ for $\beta E_b=0.47(3)$ (purple circles), $\beta E_b=0.26(2)$ (green triangles), $\beta E_b=0.06(1)$ (blue daimonds), and $\beta E_b=0.005(1)$ (red squares). Inset in (a) shows a zoomed in view of $F$ at low temperature. Dashed lines show the calculated energies using the virial expansion for the same values of $\beta E_b$ and grey lines show the ideal gas result. (c) Normalized pressure $\tilde{p}$ as a function of the interaction parameter $\ln (k_F{a}_{2\mathrm{D}})$ and temperature $T/T_F$. Solid lines show how the interaction parameter varies as a function of the relative temperature within single clouds and grey circles on the contours indicate the approximate location of the peak in $n(\mu ,T)/n_0(\mu ,T)$.