TY - JOUR
T1 - On the low density regime of homogeneous electron gas
AU - Banks, Tom
AU - Zhang, Bingnan
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2020/1
Y1 - 2020/1
N2 - We investigate the low density limit of the Homogeneous Electron system, often called the Strictly Correlated regime. We begin with a systematic presentation of the expansion around infinite rS, based on the first quantized treatments suggested in the existing literature. We show that the expansion is asymptotic in the parameter rS 1∕4 and that the leading order result contains exponential corrections that are significant even for rS∼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 (Schulz, 1993 [1]) 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.
AB - We investigate the low density limit of the Homogeneous Electron system, often called the Strictly Correlated regime. We begin with a systematic presentation of the expansion around infinite rS, based on the first quantized treatments suggested in the existing literature. We show that the expansion is asymptotic in the parameter rS 1∕4 and that the leading order result contains exponential corrections that are significant even for rS∼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 (Schulz, 1993 [1]) 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.
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U2 - 10.1016/j.aop.2019.168019
DO - 10.1016/j.aop.2019.168019
M3 - Article
AN - SCOPUS:85075087039
SN - 0003-4916
VL - 412
JO - Annals of Physics
JF - Annals of Physics
M1 - 168019
ER -