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

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 ($0.01\le\Theta= k_B T/E_F \le 10$). Within the self-consistent dielectric formalism, we construct two-dimensional versions of the Singwi-Tosi-Land-Sj\"olander (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 $(r_s,\Theta)$ 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.