TY - JOUR

T1 - On the low density regime of homogeneous electron gas

AU - Banks, Tom

AU - Zhang, Bingnan

N1 - Funding Information:
This work is partially supported by the U.S. Department of Energy and TB would like to thank N. Andrei, E. Rabinovici, S. Sachdev, D. Vanderbilt, E. Lieb, P. Gori Giorgi, K. Burke, M. Fisher, S. Kivelson, B. Spivak, and especially K. Haule for crucial insights and helpful suggestions, and the entire condensed matter group at Rutgers university for patiently listening to an interloper on their territory and making numerous helpful comments. Appendix
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

VL - 412

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

M1 - 168019

ER -