Exact ground-state properties of one-dimensional electron gas at high density

TL;DR

This paper derives exact analytical expressions for the ground-state exchange and correlation energies of a one-dimensional electron gas at high density, providing insights into the Coulomb limit and regularization independence.

Contribution

It presents the first exact analytical formulas for exchange, structure factor, and correlation energy of 1D electron gases at high density, including regularization independence.

Findings

Exact exchange energy expressions for arbitrary polarization and width.

Analytical structure factor and correlation energy at high density.

Validation of small $r_s$ expansion for distances less than 1.15.

Abstract

The dynamical response theory is used to obtain an analytical expression for the exchange energy of a quantum wire for arbitrary polarization and width. It reproduces the known form of exchange energy for 1D electron gas in the limit of infinitely thin cylindrical and harmonic wires. The structure factor for these wires are also obtained analytically in the high-density or small $r_s$ limit. This structure factor enables us to get the {\it exact} correlation energy for both the wires and demonstrates that there are at least two methods to get the ideal Coulomb limit in one dimension. The structure factor and the correlation energy are found to be independent of the way the one-dimensional Coulomb potential is regularized. The analytical expression for the pair correlation function is also presented for small distances and provides a justification for the small $r_s$ expansion as long as $r_s< \frac{3}{2} \left(\frac{\pi^2}{\pi^2+3} \right)=1.15$.