Dynamic Response of an Electron Gas: Towards the Exact Exchange-Correlation Kernel

Precise calculations of dynamics in the homogeneous electron gas (jellium model) are of fundamental importance for design and characterization of new materials. We introduce a diagrammatic Monte Carlo technique based on algorithmic Matsubara integration that allows us to compute frequency and momentum resolved finite temperature response directly in the real frequency domain using a series of connected Feynman diagrams. The data for charge response at moderate electron density are used to extract the frequency dependence of the exchange-correlation kernel at finite momenta and temperature. These results are as important for development of the time-dependent density functional theory for materials dynamics as ground state energies are for the density functional theory.

Figure 1

Real and imaginary parts of the dielectric function as functions of frequency at different momenta and temperatures for $r_s=1$ (a),(b) and $r_s=2$ (c),(d). Solid curves with symbols: ADiagMC results. Dotted curves: RPA results for the same parameter sets including $T$ and $\eta$ values. Insets in (a) and (b) present results for larger momentum transfer $Q$. Insets in (c) and (d) show the effect of lowering the temperature $T$. Errors are within the symbol sizes.

Figure 2

Loss function $\mathrm{Im}{\epsilon }^{-1}$ at $T/{ϵ}_F=0.1$ for $r_s=1$ (a), $r_s=2$ (b), and $r_s=3$ (c). The plasmon dispersion in RPA is shown by small black circles. For this data set $\eta ={ϵ}_F/20$.

Figure 3

Real (a) and imaginary (b) parts of $\chi$ as functions of $\mathrm{\Omega }$ at $T/{ϵ}_F=0.1$ for $r_s=2$ and momenta $Q/k_F=0.20027$, 0.50011. Simulation results are shown with red and blue curves with symbols. Black dotted curves: $\eta$-dependent RPA results for $Q/k_F=0.50011$ truncated at the figure scale. Errors are within the symbol sizes.

Figure 4

Real (a),(b),(c) and imaginary (d),(e),(f) parts of the exchange-correlation kernel $K_{\mathrm{xc}}$ in jellium as functions of frequency at $T/{ϵ}_F=0.1$, $r_s=1$, and 2, and several values of momentum $Q$.

Figure 5

Real parts of the exchange-correlation kernel $K_{\mathrm{xc}}(Q=0,\mathrm{\Omega }=0)$ at $T/{ϵ}_F=0.1$ as functions of the inverse diagrammatic order $N$ for $r_s=1$ (triangles) and $r_s=2$ (diamonds). Exponential fits $a+b{e}^{-cN}$ (black dotted lines) were used to perform extrapolation towards an infinite diagrammatic order limit, shown by blue symbols. Red stars: static $\mathrm{Re}{K}_{\mathrm{xc}}(Q=0)$ from [28] (in our units).