On the Low Density Regime of Homogeneous Electron Gas

TL;DR

This paper analyzes the low density limit of the homogeneous electron gas, revealing the asymptotic nature of the expansion, the absence of Wigner crystallization in one dimension, and proposing a renormalization group approach for better understanding.

Contribution

It introduces a new renormalization group scheme to study the low density regime, addressing limitations of previous asymptotic expansions and clarifying the absence of Wigner crystals in one dimension.

Findings

Expansion around infinite $r_S$ is asymptotic and limited in utility.

No Wigner Crystal exists in one spatial dimension due to Mermin-Wagner theorem.

A new RG-based approximation scheme simplifies the low density analysis.

Abstract

We investigate the low density limit of the Homogeneous Electron system, often called the {\it Strictly Correlated} regime. We begin with a systematic presentation of the expansion around infinite $r_S$, based on the first quantized treatments suggested in the existing literature. We show that the expansion is asymptotic in the parameter $r_S^{1/4}$ and that the leading order result contains exponential corrections that are significant even for $r_S \sim 100$. Thus, the systematic expansion is of limited utility. As a byproduct of this analysis, we find that there is no Wigner Crystal (WC) in one spatial dimension. This is an example of the Mermin-Wagner theorem, but was not appreciated in some earlier literature. More modern work has come to conclusions identical to ours. Note that the long range Coulomb potential modifies the dispersion relation of phonons in one dimension, but still leads to the instability of the crystal, due to a very weak infrared divergence. We then propose a new approximation scheme based on renormalization group ideas. We show that the Wegner-Houghton-Wilson-Polchinski exact renormalization group equation reduces, in the low density limit, to a classical equation for scale dependent electron and plasmon fields. In principle, this should allow us to lower the wave number cutoff of the model to a point where Wigner's intuitive argument for dominance of the classical Coulomb forces becomes rigorously correct.