Abstract
Annals of Physics 412 (2020): 168019 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.