Dielectric formalism of the 2D uniform electron gas at finite temperatures

TL;DR

This work extends the self-consistent dielectric formalism to the finite-temperature two-dimensional electron gas (2D-UEG) by implementing STLS and HNC closures and benchmarking them against new path-integral Monte Carlo data over a broad $(r_s,\Theta)$ domain. It delivers detailed analyses of the static structure factor $S(\mathbf{q})$, the static density response $\chi(\mathbf{q})$, and thermodynamic quantities, and provides an accurate, globally valid parametrization of the exchange–correlation free energy $f_{xc}(r_s,\Theta)$ (with high-density HF consistency). The results identify regimes where STLS/HNC are reliable and highlight dimension-specific behavior such as the 2D plasmon screening and the strong coupling-induced correlation peak around $q\approx 2.5 q_F$. This dataset furnishes a practical finite-temperature reference for 2D Coulomb systems and supports finite-temperature density-functional theory, while outlining paths for advancing dielectric schemes and dynamic properties in two dimensions.

Abstract

We present a comprehensive analysis of the two-dimensional uniform electron gas (2D-UEG or more commonly 2DEG) at finite temperature, spanning a broad range of densities / coupling strengths ($0.01\le{r}_s\le20$) and temperatures / degeneracy parameters ($0.01\leΘ= k_B T/E_F \le 10$). Within the self-consistent dielectric formalism, we construct two-dimensional versions of the Singwi-Tosi-Land-Sjölander (STLS) and hypernetted-chain (HNC) approximation based schemes. We benchmark the accuracy of the STLS and the HNC schemes against new state-of-the-art path-integral Monte Carlo data. We also report structural and thermodynamic properties across the full $(r_s,Θ)$ phase diagram domain studied, identify regimes in which these schemes remain quantitatively reliable, and provide an accurate parametrization of the exchange--correlation free energy of the finite-temperature 2DEG.

Figures (9)

• Figure 1: Average sign $S$ of the 2DEG as a function of particle number $N$ at $\Theta=4$ and various values of the coupling parameter $r_s$.
• Figure 2: Static structure factor $S(\mathbf{q})$ of the 2DEG at $r_s=1$ (left), $r_s=4$ (center), and $r_s=10$ (right) with the top and bottom rows corresponding to $\Theta=4$ and $\Theta=1$, respectively. Black symbols: quasi-exact PIMC reference data; solid green: HNC scheme; dashed red: STLS scheme; dotted yellow: RPA. The dotted gray lines correspond to the exact $q\to0$ limit, see Eq. \ref{['eq:SSF0']}).
• Figure 3: Dependence of the 2DEG static structure factor $S(\mathbf{q})$ on the coupling parameter $r_s$ at $\Theta=4$ (left) and $\Theta=1$ (right). The stars and crosses show quasi-exact PIMC reference results for different particle numbers $N$, while the gray, yellow, green, red, and blue symbols correspond to $r_s=20, 10, 5, 1, 0.5$, respectively. The STLS results are included as dash-dotted lines with the same color code.
• Figure 4: Parametric sweep for the STLS generated 2DEG static structure factor $S(\mathbf{q})$. Left: dependence of $S(\mathbf{q})$ on the coupling parameter $r_s=0.1,1,2,5,10,20$ at $\Theta=1.0$. Right: dependence of $S(\mathbf{q})$ on the degeneracy parameter $\Theta=0.01,0.1,1,2,5,10$ at $r_s=1.0$.
• Figure 5: Dependence of the 2DEG static structure factor $S(\mathbf{q})$ on the degeneracy temperature $\Theta=0.5,\,0.75,\,1.0,\,2.0,\,4.0$ at $r_s=10$. The colored symbols and lines correspond to the PIMC data (for $N=14,\,34$) and STLS results, respectively.