Permutation-blocking path-integral Monte Carlo approach to the static density response of the warm dense electron gas

The static density response of the uniform electron gas is of fundamental importance for numerous applications. Here we employ the recently developed ab initio permutation blocking path integral Monte Carlo (PB-PIMC) technique [T. Dornheim et al., New J. Phys. 17, 073017 (2015)] to carry out extensive simulations of the harmonically perturbed electron gas at warm dense matter conditions. In particular, we investigate in detail the validity of linear response theory and demonstrate that PB-PIMC allows us to obtain highly accurate results for the static density response function and, thus, the static local field correction. A comparison with dielectric approximations to our new ab initio data reveals the need for an exact treatment of correlations. Finally, we consider a superposition of multiple perturbations and discuss the implications for the calculation of the static response function.

Figure 1

Screen shots of standard path integral Monte Carlo simulations of the warm dense UEG for $N=19$ spin-polarized electrons, $r_s=1$, and $P=32$, with $\theta =8$ (a), $\theta =1$ (b), and $\theta =0.3$ (c).

Figure 2

Screen shot of a permutation blocking path integral Monte Carlo simulation of the UEG with $N=9$ spin-polarized electrons with $r_s=1, \theta =1$, and $P=2$ imaginary time propagators. The green, blue, and purple points correspond to the three different kinds of time slices, see Refs. [68, 69, 70].

Figure 3

Density profiles along the $x$ direction for $N=54, r_s=10$, and $\theta =1$. Shown are PB-PIMC results for $P=4$ with $\mathbf{q}=2\pi L^{-1}{(2,0,0)}^T$ and weak (a), medium (b), and strong (c) perturbations. The black lines correspond to fits according to Eq. (34).

Figure 4

Induced density modulation for $N=54, r_s=10$, and $\theta =1$. Shown are PB-PIMC results for $P=4$ with $\mathbf{q}=2\pi L^{-1}{(q_x,0,0)}^T [q_x=2$ (a) and $q_x=1$ (b)] directly computed from QMC, cf. Eq. (32), and from fits according to Eq. (34).

Figure 5

Average sign for $N=54, r_s=10$, and $\theta =1$ (a). Shown are PB-PIMC results for $P=4$ with $\mathbf{q}=2\pi L^{-1}{(q_x,0,0)}^T$. Corresponding density profiles along the $x$ direction for $A=0.1$ (b).

Figure 6

Convergence with number of propagators $P$ for $N=34, r_s=10$, and $\theta =1$ with a perturbation of wave vector $\mathbf{q}=2\pi L^{-1}{(1,0,0)}^T$ and amplitude $A=0.01$. Shown are QMC results for the density matrix (a) and the density profile along the $x$ direction (b).

Figure 7

Convergence with number of propagators $P$ for $N=54, r_s=10$, and $\theta =1$ with the perturbation of wave vector $\mathbf{q}=2\pi L^{-1}{(5,0,0)}^T$ and amplitude $A=0.01$. Shown are QMC results for the density matrix (a) and the potential energy, i.e., the sum of Ewald interaction and external field (b).

Figure 8

Wave-vector dependence of the static response function for the unpolarized UEG at $r_s=10$ and $\theta =1$. Shown are QMC results according to Eq. (33) for different particle numbers (symbols) and the predictions from RPA (gray) and STLS (red). The black arrow indicates the Fermi wave vector, $k_F={(9\pi /4)}^{1/3}/r_s$. Panel (b) shows a magnified segment.

Figure 9

Wave-vector dependence of the static response function for the unpolarized UEG at $r_s=10$ and $\theta =4$. Shown are QMC results according to Eq. (33) for $N=8$ electrons obtained from PB-PIMC with $P=4$ (black squares) and standard PIMC with $P=100$ (green crosses). As a reference, we also show the predictions from RPA (gray) and STLS (red).

Figure 10

Density profile along $x$ direction for $N=54, r_s=10$, and $\theta =1$ with a perturbation amplitude of $A=0.005$. The green squares correspond to a QMC simulation with a superposition of two $\mathbf{q}$ vectors ($q_x=1$ and $q_x=2$), see Eq. (37), whereas the yellow and red points have been obtained using two separate QMC simulations each with a single perturbation. The black crosses correspond to a superposition of the latter two. The blue lines have been reconstructed from a fit to the green squares according to Eq. (38), i.e., by obtaining both $\chi \left({\mathbf{q}}_1\right)$ and $\chi \left({\mathbf{q}}_2\right)$ from the density response of the system with two simultaneous perturbations.

Figure 11

Induced density for $N=54, r_s=10$, and $\theta =1$ for a perturbation of wave vector $\mathbf{q}=2\pi L^{-1}{(q_x,0,0)}^T$. The blue crosses have been obtained from a QMC simulation with a single perturbation, whereas the green squares and red circles correspond to the direct and cosine-fit results from the simulation with a double perturbation. Finally, the black lines has been obtained by a linear fit to the green squares.

Figure 12

Perturbation strength dependence for a combination of three wave vectors ${\mathbf{q}}_i=2\pi L^{-1}(q_{x,i},0,0)$ with $q_{x,1}=1, q_{x,2}=2$, and $q_{x,3}=3$. The black squares correspond to direct QMC results according to Eq. (32), the green crosses to direct QMC results from a simulation with a single perturbation, and the red line to a fit in the linear response regime.