Dielectric formalism of the two-dimensional 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 ≤ r s ≤ 20 ) and temperatures/degeneracy parameters ( 0.01 ≤ Θ = k B T / E F ≤ 10 ) . Within the self-consistent dielectric formalism, we construct 2D 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.