Universal Equation of State and Pseudogap in the Two-Dimensional Fermi Gas

We determine the thermodynamic properties and the spectral function for a homogeneous two-dimensional Fermi gas in the normal state using the Luttinger-Ward, or self-consistent $T$-matrix, approach. The density equation of state deviates strongly from that of the ideal Fermi gas even for moderate interactions, and our calculations suggest that temperature has a pronounced effect on the pressure in the crossover from weak to strong coupling, consistent with recent experiments. We also compute the superfluid transition temperature for a finite system in the crossover region. There is a pronounced pseudogap regime above the transition temperature: the spectral function shows a Bogoliubov-like dispersion with backbending, and the density of states is significantly suppressed near the chemical potential. The contact density at low temperatures increases with interaction and compares well with both experiment and zero-temperature Monte Carlo results.

Figure 1

Density of states $\rho (\omega )$, normalized by ${\rho }_0=m/2\pi$ for the free Fermi gas, at interaction $\ln (k_F{a}_{2\mathrm{D}})=0.8$ for different temperatures: $T=0.45T_F$ (top curve at $\omega =0$) to $T=0.07T_F$ (bottom). Inset: Spectral function $A(k,\omega )$ for $T=0.07T_F$. The grey dashed line marks the maximum in the spectral weight of the bottom band.

Figure 2

Density $n$ of the 2D Fermi gas vs chemical potential $\beta \mu$, for different interaction strengths $\beta {ϵ}_B$ (see legend). Since the density is normalized by $n_0(\beta \mu )$ for the noninteracting gas, the nonmonotonic behavior of $n/n_0$ reflects the impact of interactions, while the compressibility $\kappa =(\partial n/\partial \mu )/n^2$ is always positive. The inset shows a typical trajectory in $T/T_F$ vs $\ln (k_F{a}_{2\mathrm{D}})$ corresponding to the dotted line of fixed $\beta {ϵ}_B=1$. Along this line, $\beta \mu$ increases with decreasing $T/T_F$.

Figure 3

Pressure $P$ vs interaction strength, normalized by the pressure $P_0=n{ϵ}_F/2$ of an ideal Fermi gas of the same density at $T=0$. Luttinger-Ward data at temperature $T/T_F=0.2$ (top, dotted line) to $T/T_F=0.1$ (solid line) in comparison with experimental data [10] (symbols) and $T=0$ QMC results [11] (dashed line).

Figure 4

Critical temperature $T_c/T_F$ vs interaction strength $\ln (k_F{a}_{2\mathrm{D}})$. The Luttinger-Ward result for a finite system (blue solid line) in the crossover region $\ln (k_F{a}_{2\mathrm{D}})\gtrsim 0$ is compared with analytical limits [12]. The red dots mark the crossover temperature $T^{*}$ to the pseudogap regime for $\ln (k_F{a}_{2\mathrm{D}})\gtrsim 1$.

Figure 5

Contact density $C$ vs interaction strength $1/\ln (k_F{a}_{2\mathrm{D}})$ at temperature $T/T_F=0.27$. We compare our result (blue solid line) with the experimental data at $T/T_F=0.27$ [13] (red symbols), the weak-coupling result (green dashed-dotted line), and QMC calculations at $T=0$ [11] (cyan dashed line).