A unified framework for the Kondo problem and for an impurity in a Luttinger liquid

TL;DR

This paper presents a unified theoretical framework for the anisotropic Kondo model and boundary sine-Gordon model, enabling exact results for impurity problems in quantum field theories and Luttinger liquids.

Contribution

It unifies two boundary integrable models using quantum group representations, providing new exact results and methods for analyzing impurity effects.

Findings

Exact coefficients for the anisotropic Kondo model in magnetic fields.

Continuation of results from anisotropic to isotropic regimes.

Non-equilibrium conductance for impurities in Luttinger liquids.

Abstract

We develop a unified theoretical framework for the anisotropic Kondo model and the boundary sine-Gordon model. They are both boundary integrable quantum field theories with a quantum-group spin at the boundary which takes values, respectively, in standard or cyclic representations of the quantum group $SU(2)_q$. This unification is powerful, and allows us to find new results for both models. For the anisotropic Kondo problem, we find exact expressions (in the presence of a magnetic field) for all the coefficients in the ``Anderson-Yuval'' perturbative expansion. Our expressions hold initially in the very anisotropic regime, but we show how to continue them beyond the Toulouse point all the way to the isotropic point using an analog of dimensional regularization. For the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid, we find the non-equilibrium conductance for all values of the Luttinger coupling.