The on-shell self-energy of the uniform electron gas in its weak-correlation limit

TL;DR

This paper revisits the RPA treatment of the uniform electron gas's ground-state energy in the weak-correlation limit, focusing on the self-energy's consistency with fundamental theorems and sum rules.

Contribution

It analyzes the agreement of different self-energy treatments with the Hugenholtz-van Hove theorem in the weak-correlation limit.

Findings

RPA asymptotics include a logarithmic term in r_s

Renormalized RPA diagrams yield consistent asymptotic expressions

Sum rules connect exchange contributions in the second order

Abstract

The ring-diagram partial summation (or RPA) for the ground-state energy of the uniform electron gas (with the density parameter $r_s$) in its weak-correlation limit $r_s\to 0 $ is revisited. It is studied, which treatment of the self-energy $\Sigma(k,\omega)$ is in agreement with the Hugenholtz-van Hove (Luttinger-Ward) theorem $\mu-\mu_0= \Sigma(k_{\rm F},\mu)$ and which is not. The correlation part of the lhs h as the RPA asymptotics $a\ln r_s +a'+O(r_s)$ [in atomic units]. The use of renormalized RPA diagrams for the rhs yields the similar expression $a\ln r_s+a''+O(r_s)$ with the sum rule $a'= a''$ resulting from three sum rules for the components of $a'$ and $a''$. This includes in the second order of exchange the sum rule $\mu_{2{\rm x}}=\Sigma_{2{\rm x}}$ [P. Ziesche, Ann. Phys. (Leipzig), 2006].