Momentum distribution function and short-range correlations of the warm dense electron gas: Ab initio quantum Monte Carlo results

In a classical plasma the momentum distribution, $n\left(k\right)$, decays exponentially, for large $k$, and the same is observed for an ideal Fermi gas. However, when quantum and correlation effects are relevant simultaneously, an algebraic decay, $n_{\infty }\left(k\right)\sim k^{-8}$ has been predicted. This is of relevance for cross sections and threshold processes in dense plasmas that depend on the number of energetic particles. Here we present extensive ab initio results for the momentum distribution of the nonideal uniform electron gas at warm dense matter conditions. Our results are based on first principle fermionic path integral Monte Carlo (CPIMC) simulations and clearly confirm the $k^{-8}$ asymptotic. This asymptotic behavior is directly linked to short-range correlations which are analyzed via the on-top pair distribution function (on-top PDF), i.e., the PDF of electrons with opposite spin. We present extensive results for the density and temperature dependence of the on-top PDF and for the momentum distribution in the entire momentum range.

Figure 1

Illustration of a path $C$, Eq. (18), with five kinks. The three kinks 1, 3, 5, at times $t_1, t_3, t_5$, each involve four orbitals: ${\kappa }_1=(1,4;3,5), {\kappa }_3=(1,3;0,2), {\kappa }_5=(4,5;1,3)$ respectively. The two kinks 2 and 4, at $t_2$ and $t_4$, involve two orbitals each: ${\kappa }_2=(0;1)$ and ${\kappa }_4=(2;4)$. The fourth Slater determinant $|n^{\left(4\right)}\rangle$ exists between the imaginary “times” $t_3$ and $t_4$ and contains three occupied orbitals $\left\{0,2,5\right\}$.

Figure 2

Temperature dependence of the momentum distribution of moderately correlated electrons, $r_{\mathrm{s}}=0.5$. CPIMC results with $N=54$ particles for three temperatures are compared to the ground state (solid black, data of Ref. [84]). For comparison, the ideal Fermi distribution is shown by dashed lines of the same color as the interacting result (from left to right $\mathrm{\Theta }=1.5,2.0,4.0$).

Figure 3

Density dependence of the momentum distribution of moderately correlated electrons at temperature $\mathrm{\Theta }=2$. CPIMC results with $N=54$ particles are compared to the ideal Fermi-Dirac distribution $n^{\mathrm{id}}$ (full black line). For momenta below approximately $6k_F$ the correlated distributions are indistinguishable from $n^{\mathrm{id}}$. For comparison, the ground state distributions, as given by Ref. [84], are shown by the dashed lines of the same color as the finite temperature result (from top to bottom: $r_s=0.7,0.2$).

Figure 4

Upper panel: deviation of the momentum distribution (CPIMC results with $N=54$ particles) from the ideal Fermi-Dirac distribution at moderate coupling $r_{\mathrm{s}}=0.5$. Lower panel: Difference of the distribution functions weighted with $k^4/2$ (Hartree units), i.e., k-resolved kinetic energy density.

Figure 5

Same as Fig. 4, but for a fixed temperature, $\mathrm{\Theta }=2$, and three densities. The different ordering of the curves in the lower panel arises from the $r_s$-dependence of the horizontal scale, $k_F\propto r_s^{-1}$.

Figure 6

Interaction-induced lowering of the kinetic energy. Heat map: $K_{x}c$, Eq. (36), computed from the parametrization by Groth et al. [80]. Solid black line: $r_s-\mathrm{\Theta }$-combinations where $K_{x}c$ vanishes; dotted lines: uncertainty interval of $5\times {10}^{-3}\mathrm{Ha}$. Solid blue line: $K_{x}c=0$ according to RPIMC results of Ref. [68]. Red (green) pluses: CPIMC results for kinetic energy decrease (increase) compared to ideal case. Red circles (green crosses): CPIMC data points where the occupation of the lowest orbital, $n\left(0\right)$, is higher (lower) than in the ideal case, i.e., $n^{\mathrm{id}}\left(0\right)$. Extensive data for the kinetic energy are presented in the tables in Appendix pp2.

Figure 7

Large momentum behavior of the momentum distribution, for $r_{\mathrm{s}}=0.2$ and $\mathrm{\Theta }=0.0625$. Top: blue line. CPIMC results for $N=54$ particles, pink line: best fit to the asymptotic, Eq. (3), with $g_{\uparrow \downarrow }\left(0\right)$ taken from CPIMC data; green: ground state value from Ref. [96]. Bottom: relative difference of CPIMC and the ground state data from the asymptotic (pink line in top plot), according to Eq. (37).

Figure 8

Same as Fig. 7, but for $r_s=0.5$ and $\mathrm{\Theta }=2.$

Figure 9

On-top PDF, $g_{\uparrow \downarrow }\left(0\right)=2g\left(0\right)$, for $N=54$ electrons at $r_s=1$ and $\theta =4$, from different methods. Red circles: CPIMC results for $g_{\uparrow \downarrow }\left(0\right)$ for different values of the momentum cutoff, $E_{m}ax$ (top $x$-axis); solid red line: linear fit. Green crosses: standard PIMC data for the distance-dependent PDF, $g_{\uparrow \downarrow }\left(r\right)$, (bottom $x$-axis); solid green curve: linear fit. Blue diamonds: RPIMC data from Ref. [71] for the same conditions but $N=66$.

Figure 10

Influence of the particle number on the on-top PDF at $\mathrm{\Theta }=0.0625$ for CPIMC results with $N=66, N=54$, and 38 particles. We show the relative deviation of each respective data point from the corresponding CPIMC result for $N=14$, which corresponds to the horizontal line at 0. Simulation results from the approximate RCPIMC and $\text{RCPIMC}+$ methods [82] are included. “TA” denotes twist angle averaging.

Figure 11

Same as Fig. 10, but for the temperature $\mathrm{\Theta }=2$.

Figure 12

Temperature dependence of the on-top pair distribution for $r_{\mathrm{s}}=0.2$ from CPIMC simulations with $N=54$ particles. Very good agreement of $\text{RCPIMC}+$ [82] with CPIMC is confirmed.

Figure 13

Temperature dependence of the on-top PDF for $r_{\mathrm{s}}=0.5$, from CPIMC simulations with 14 particles. Twist angle averaging has been applied. Shaded area indicates the statistical error. The minimum temperature is set by the fermion sign problem. For better visibility, the curves for $r_s=0.5$ and 0.7 are shifted vertically by the number given in parentheses.

Figure 14

Analysis of the minimum of the on-top PDF. Filled symbols correspond to the location of the minimum in the $\mathrm{\Theta }\text{−}{r}_{\mathrm{s}}$ plane (left axis $\mathrm{\Theta }$). Open symbols correspond to the minimum value of the OT-PDF (right axis). Orange circles: CPIMC results for $N=14$ particles. The green line represents the values of $g\left(0\right)$ at a fixed temperature $\mathrm{\Theta }=0.656$. ESA: results of the extended static approximation [58]; see text.

Figure 15

Density dependence of the on-top PDF for $\mathrm{\Theta }=1/16$ at weak coupling. Open (filled) circles: CPIMC ($\text{RCPIMC}+$) results for $N=54$ particles. Lines: results of ground state models, i.e., Eq. (9) (Overhauser model) and of Calmels et al. [100].

Figure 16

Density dependence of the on-top PDF for $\mathrm{\Theta }=1$. Red squares: CPIMC data [$r_s\le 0.4$: $N=54$ particles; $r_s\ge 0.5$: $N=14$]. Blue circles: FP-PIMC data [$0.6<r_s\le 8$: $N=66$; $0.1\le r_s\le 0.6$: $N=34$]. Black full (dashed) lines: ground state DMC simulations [67] and Eq. (9), respectively. Inset: zoom into the high-density range (linear scale).

Figure 17

Top: On-top PDF (top), bottom: the function (43) for two temperatures: $\mathrm{\Theta }=4$ (orange line and symbols) and $\mathrm{\Theta }=2$ (green line and symbols). Triangles: CPIMC data for $N=54$; squares: FP-PIMC with $N=66$ particles. Dotted lines: parametrization of Dornheim et al. [58]; black line: ground state parametrization of Calmels et al. Note the extended $r_s$-range in the lower figure. For more data on the maximum of $s\left(r_s\right)$, see Table 1.

Figure 18

Illustration of the prescription (44) to determine the onset of the large-momentum asymptotic from the intersection of the ideal Fermi function, $f^{\mathrm{id}}\left(k\right)$, (dashes), with the $k^{-8}$ asymptotic, $n_{\infty }\left(k\right)$, (full lines of the same color). The asymptotic is determined from CPIMC simulations of the on-top PDF for $N=54$ particles.

Figure 19

Onset $k_{\infty }$ of the large-momentum asymptotic, as calculated from Eq. (44). The procedure is illustrated in Fig. 18. CPIMC simulations with $N=14$ particles. The lower limit of $\mathrm{\Theta }$, for the different curves, is set by the fermion sign problem.