Tunneling in quantum wires I: Exact solution of the spin isotropic case

TL;DR

This paper presents an exact solution for impurity tunneling in a spin isotropic Luttinger liquid, revealing the renormalization group flow, conductance behavior, and a duality between high and low energy regimes.

Contribution

It introduces an exactly solvable model for impurity tunneling in spinful Luttinger liquids with a novel impurity representation linked to quantum algebra.

Findings

RG flow analysis shows IR fixed point with two separate leads

Exact DC conductance computed using non-perturbative methods

Identifies a duality between UV and IR current expansions

Abstract

We show that the problem of impurity tunneling in a Luttinger liquid of electrons with spin is solvable in the spin isotropic case ($g_\sigma=2$, $g_\rho$ arbitrary). The resulting integrable model is similar to a two channel anisotropic Kondo model, but with the impurity spin in a "cyclic representation" of the quantum algebra $su(2)_q$ associated with the anisotropy. Using exact, non-perturbative techniques we study the RG flow, and compute the DC conductance. As expected from the analysis of Kane and Fisher we find that the IR fixed point corresponds to two separate leads. We also prove an exact duality between the UV and IR expansions of the current at vanishing temperature.