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

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 $V(x)=\frac{e^2{d}^{-ϵ}}{\mathrm{\mid }x{\mathrm{\mid }}^{1-ϵ}}$. As a consistency check of our computational technique we also consider the particular case of a Coulomb potential. Our result confirms the $\exp -C(logx{)}^{3\mathrm{/}2}$ long-distance behavior first obtained by Schulz.