The basic characters of the electronic correlation effect from weak to strong in the three dimensional electron gas

TL;DR

This paper uses eigenfunctional theory to accurately compute the ground state energy and phase transition of the three-dimensional electron gas, especially at low densities where previous models fail.

Contribution

It introduces an eigenfunctional approach that ensures positive pair distribution functions and accurately predicts a continuous paramagnetic to ferromagnetic phase transition.

Findings

Eigenfunctional theory yields positive g(r) at all densities.

Identifies a phase transition at r_s=19.9±0.8.

Results align with experimental observations and previous studies.

Abstract

In this paper, we present the rigorous expression of the ground state energy and study the phase transition of the three-dimensional homogeneous electron gas by the eigenfunctional theory. The ground state energy is decided completely by the pair distribution function $ g(r)$, which is, by definition, strictly a positive function. But when the density decreases, the electronic correction effect becomes strong from weak, the previous theories basing on one-particle approximation, such as: RPA, Hubbard and STLS ,\cite{4,9} can't insure $g(r)$ positive always, which implies that they are becoming invalid and overestimate the ground state energy. The eigenfunctional theory has a significant improvement over them, under the linear approximation in solving the equation of the phase field, the $g(r)$ obtained by the eigenfunctional theory is always positive and can well satisfy the normalization integral ($n_{0}\int d^3r[g(r)-1]=-1$) which is one of the important features of $g(r)$. After obtaining $g(r)$ and calculating the ground state energy of the electron gas both in paramagnetic and ferromagnetic by $g(r)$, we observe a continuous phase transition from paramagnetic to the ferromagnetic occurring at $r_s=19.9\pm0.8$. This can be indirectly supported by the observation of a ferromagnetic state in doped hexaboride $(Ca_{1-x}La_xB_b)$,\cite{42} and is close to the result of G.Ortiz et.al ($r_s=20\pm5$).\cite{12}