Exchange-correlation effect in the charge response of a warm dense electron gas

The study of warm dense matter, widely existing in nature and laboratories, is challenging due to the interplay of quantum and classical fluctuations. We develop a variational diagrammatic Monte Carlo method and determine the exchange-correlation kernel $K_{\mathrm{XC}}(q;T)$ of a warm dense electron gas for a broad range of temperature $T$ and momentum $q$. We observe several interesting physics, including the $T$-dependent evolution of the hump structure and the large-$q$ tail and the emergence of a scaling relation. Particularly, by deriving an analytical form for $q\to \infty$, we obtain large-$q$ tails of $K_{\mathrm{XC}}$ with high precision. It is shown that the $K_{\mathrm{XC}}$ data can be reliably transformed into real space, which can be directly used in density-functional-theory calculations of real warm dense matter.

Figure 1

Static local field correction $G\left(q\right)$ in the extensive ($q,T$) plane for $r_s=1$. The circles depict VDMC data points for $\theta =0.0625,0.25,0.4,1,2,4,8$. The cyan squares and dotted line depict the tendency of the maximum momentum $q_{\max }$ of $G\left(q\right)$ versus $T$.

Figure 2

(a) Rescaled static LFC as a function of $q/\sqrt{T}$. The data points with solid lines are for $r_s=1$ and $\theta =0.25,0.4,1,2,4,8$, and ones with dashed lines are for $r_s=2$ and $\theta =4,8$. As $T\to \infty$, the LFC data tend to collapse to a universal curve that depends on $q/\sqrt{T}$, and is covered in the pink shadow. The inset shows the evolution of $q_{\max }$ from $\simeq 2k_{\text{F}}$ in the zero-$T$ limit to $\approx 7.6{\lambda }_{\mathrm{th}}$ in the high-$T$ limit. (b) The XC kernel $K_{\mathrm{XC}}\left(q\right)$ for different temperatures and $r_s=1$. The tail of either curve converges to a constant $K_{\infty }$ that is explicitly calculated by the VDMC as Eq. (3), and the strip marks its value and error. The inset shows the evolution of $K_{\mathrm{XC}}(q=0,T)$. Our data are consistent with the parametrized function [56] (blue line). (c) The XC kernel of UEG in real space with the delta-peaked term subtracted. The inset shows the interaction-induced shift of the kinetic energy versus temperature. Our data is approaching the high-$T$ approximation [57] (blue line) for $\theta >1$.

Figure 3

(a) Static polarization scaled by the density of states at the Fermi level $N_{\text{F}}$ at $q=3k_{\text{F}}$ versus truncation order $N$ for $\theta =2$ and $r_s=1$. All $\lambda$ choices lead to the same extrapolated value, and the optimal $\lambda$ for the fastest convergence is about $1.0$ in this case. (b) LFC $G\left(q\right)$ at $\theta =2$ and $r_s=1$ for the VDMC (red circles), the PIMC [46] (blue squares), and the finite-temperature version [96] of the STLS approximation [97] (black line). The STLS approximation is a widely used self-consistent scheme for an approximate treatment of the static LFC.