Exact electronic Green functions in a Luttinger liquid with long-range interactions

TL;DR

This paper derives an analytical expression for the electronic Green function in a Luttinger liquid with long-range interactions, confirming previous long-distance behavior predictions and extending understanding of such systems.

Contribution

The authors provide an exact analytical calculation of the Green function for a Luttinger liquid with a super long-range potential, including Coulomb interactions, validating prior asymptotic results.

Findings

Confirmed the $ ext{exp}(-C( ext{log}x)^{3/2})$ long-distance behavior

Derived explicit Green function expressions for super long-range potentials

Validated computational techniques with Coulomb potential case

Abstract

We compute the 2-point (equal-time) electronic Green function in a Tomonaga-Luttinger system with long-range electron-electron interactions. We obtain an analytical expression for a "super long-range" potential of the form $V(x)=\frac{e^2d^{-\epsilon}}{|x|^{1-\epsilon}}$. As a consistency check of our computational technique we also consider the particular case of a Coulomb potential. Our result confirms the $\exp-C(logx)^{3/2}$ long-distance behavior first obtained by Schulz.