Fermionic path-integral Monte Carlo results for the uniform electron gas at finite temperature

The uniform electron gas (UEG) at finite temperature has recently attracted substantial interest due to the experimental progress in the field of warm dense matter. To explain the experimental data, accurate theoretical models for high-density plasmas are needed that depend crucially on the quality of the thermodynamic properties of the quantum degenerate nonideal electrons and of the treatment of their interaction with the positive background. Recent fixed-node path-integral Monte Carlo (RPIMC) data are believed to be the most accurate for the UEG at finite temperature, but they become questionable at high degeneracy when the Brueckner parameter $r_s=a/a_B$—the ratio of the mean interparticle distance to the Bohr radius—approaches 1. The validity range of these simulations and their predictive capabilities for the UEG are presently unknown. This is due to the unknown quality of the used fixed nodes and of the finite-size scaling from $N=33$ simulated particles (per spin projection) to the macroscopic limit. To analyze these questions, we present alternative direct fermionic path integral Monte Carlo (DPIMC) simulations that are independent from RPIMC. Our simulations take into account quantum effects not only in the electron system but also in their interaction with the uniform positive background. Also, we use substantially larger particle numbers (up to three times more) and perform an extrapolation to the macroscopic limit. We observe very good agreement with RPIMC, for the polarized electron gas, up to moderate densities around $r_s=4$, and larger deviations for the unpolarized case, for low temperatures. For higher densities (high electron degeneracy), $r_s\lesssim 1.5$, both RPIMC and DPIMC are problematic due to the increased fermion sign problem.

Figure 1

Illustration of the extrapolation of the total energy with respect to the number of beads $M$ and the particle number $N$. First, for a fixed value of $M$, an extrapolation $N\to \infty$ is performed (upper curve). Subsequently, these results are extrapolated with respect to $M$ (lower curve). Results are for the unpolarized case with $\Theta =0.0625$ and $r_s=1$. DPIMC data points are in blue, while the extrapolation (fit) is shown by the red line.

Figure 2

Total energy per particle (left column) and correlation energy per particle (right column) for a polarized ideal $\left(E_0\right)$ and interacting electron gas (PUEG) with temperatures ranging from $\Theta =8$ to $0.0625$; see text in the graphs. Comparison of restricted PIMC (RPIMC, Ref. [24]), fermionic PIMC (SFPIMC, open pink circles [24]), and the present DPIMC results. The correlation energy is given by $E_c=E_{\text{tot}}-E_0-E_{x,\text{HF}}$ using the Hartree-Fock energy of Ref. [24]; see the main text.)

Figure 3

Same as Fig. 2 but for an unpolarized electron gas.