Ab initio quantum Monte Carlo simulations of the uniform electron gas without fixed nodes

The uniform electron gas (UEG) at finite temperature is of key relevance for many applications in the warm dense matter regime, e.g., dense plasmas and laser excited solids. Also, the quality of density functional theory calculations crucially relies on the availability of accurate data for the exchange-correlation energy. Recently, results for $N=33$ spin-polarized electrons at high density, $r_s=\overline{r}/a_B\lesssim 4$, and low temperature have been obtained with the configuration path integral Monte Carlo (CPIMC) method [T. Schoof et al., Phys. Rev. Lett. 115, 130402 (2015)]. To achieve these results, the original CPIMC algorithm [T. Schoof et al., Contrib. Plasma Phys. 51, 687 (2011)] had to be further optimized to cope with the fermion sign problem (FSP). It is the purpose of this paper to give detailed information on the manifestation of the FSP in CPIMC simulations of the UEG and to demonstrate how it can be turned into a controllable convergence problem. In addition, we present new thermodynamic results for higher temperatures. Finally, to overcome the limitations of CPIMC towards strong coupling, we invoke an independent method—the recently developed permutation blocking path integral Monte Carlo approach [T. Dornheim et al., J. Chem. Phys. 143, 204101 (2015)]. The combination of both approaches is able to yield ab initio data for the UEG over the entire density range, above a temperature of about one half of the Fermi temperature. Comparison with restricted path integral Monte Carlo data [E. W. Brown et al., Phys. Rev. Lett. 110, 146405 (2013)] allows us to quantify the systematic error arising from the free particle nodes.

Figure 1

Available ab initio quantum Monte Carlo data in the warm dense matter range for $N=33$ spin-polarized electrons. Dots: CPIMC. Squares: PB-PIMC. Red: Additional combined CPIMC and PB-PIMC results of this paper. Gray: Previous results from CPIMC [21] and PB-PIMC [25], respectively. ICF: Typical inertial confinement fusion parameters [10]. Quantum (classical) behavior dominates below (above) the line $\mathrm{\Theta }=1$. $\mathrm{\Gamma }=e^2/\overline{r}{k}_{B}T$ is the classical coupling parameter.

Figure 2

Typical closed path in Slater determinant (Fock) space. The state with three occupied orbitals $|{\vec{k}}_0\rangle ,|{\vec{k}}_1\rangle ,|{\vec{k}}_3\rangle$ undergoes a two-particle excitation $s_1$ at time ${\tau }_1$ replacing the occupied orbitals $|{\vec{k}}_0\rangle ,|{\vec{k}}_3\rangle$ by $|{\vec{k}}_2\rangle ,|{\vec{k}}_5\rangle$. Two further excitations occur at ${\tau }_2$ and ${\tau }_3$. The states at the “imaginary times” $\tau =0$ and $\tau =\beta$ coincide. All possible paths contribute to the partition function $Z$, Eq. (8). (Figure from Ref. [21].)

Figure 3

Average sign (a) and average number of kinks (b) of direct CPIMC, plotted versus the density parameter for different particle numbers in $N_B=2109$ basis functions at $\theta =0.125$.

Figure 4

Average number of kinks of direct CPIMC, plotted versus the density parameter for $N=4$ particles in $N_B=5575$ basis functions at different temperatures.

Figure 5

Convergence of the internal energy with respect to the kink potential parameter $\kappa$, using different parameters $\delta$. The system consists of $N=4$ particles in $N_B=19$ basis functions at $\theta =0.5$ and $r_s=40$ for which the energy can be computed with an exact configuration interaction (CI) method (dashed black line). Each point is the result of a whole CPIMC simulation, where integer numbers from 5 to 28 have been used for $\kappa$.

Figure 6

Convergence of the internal energy with respect to the kink potential parameter $\kappa$ and extrapolation to $1/\kappa \to 0$, corresponding to $K\to \infty$, at $\theta =1.0$. (a) $N=4$ particles and $r_s=10.0$ in $N_B=5575$ basis functions. (b) $N=33$ and $r_s=1.0$ in $N_B=4169$ basis functions. The asymptotic values (black points) are enclosed between the blue and green lines and, within error bars, coincide with the PB-PIMC result (orange points).

Figure 7

Exchange-correlation energy $E_{\mathrm{xc}}$ times $r_s$ of the $N=33$ particle spin-polarized UEG over the density parameter $r_s$ for different degeneracy parameters $\theta$. Results have been obtained by combining the CPIMC (dots) and PB-PIMC (crosses) approach taking the most accurate values of each method (connected by the solid line). In addition, RPIMC results from Ref. [17] are plotted for comparison (open circles).

Figure 8

Exchange-correlation energy $E_{xc}$ times $r_s$ of the $N=33$ particle spin-polarized UEG over the degeneracy parameter $\theta$ for different density parameters $r_s$. Shown are results from CPIMC (dots) and PB-PIMC (crosses) calculations. In addition, RPIMC results from Ref. [17] are plotted for comparison (lines with light colors and open circles, for $r_s=1$ and $r_s=4$).