Renormalization group study of the Kondo problem at a junction of several Luttinger wires

TL;DR

This paper investigates the renormalization group flows of Kondo couplings in a junction of multiple Luttinger wires with an impurity spin, revealing fixed points and conductance behaviors influenced by electron interactions and junction properties.

Contribution

It derives RG equations for Kondo couplings in a multi-wire junction with Luttinger liquids and analyzes fixed points for various S-matrices and interaction strengths, extending understanding of impurity effects.

Findings

Kondo couplings flow to large antiferromagnetic values under RG.

For Griffiths S-matrix, system approaches ferromagnetic fixed point at large distances.

Strong inter-electron interactions stabilize a multi-channel fixed point.

Abstract

We study a system consisting of a junction of N quantum wires, where the junction is characterized by a scalar S-matrix, and an impurity spin is coupled to the electrons close to the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group equations for the Kondo couplings of the spin to the electronic modes on different wires, and analyze the renormalization group flows and fixed points for different values of the initial Kondo couplings and of the junction S-matrix (such as the decoupled S-matrix and the Griffiths S-matrix). We generally find that the Kondo couplings flow towards large and antiferromagnetic values in one of two possible ways. For the Griffiths S-matrix, we study one of the strong coupling flows by a perturbative expansion in the inverse of the Kondo coupling; we find that at large distances, the system approaches the ferromagnetic fixed point of the decoupled S-matrix. For the decoupled S-matrix with antiferromagnetic Kondo couplings and weak inter-electron interactions, the flows are to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. But strong inter-electron interactions, with K_\rho < N/(N+2), stabilize a multi-channel fixed point in which the coupling between different channels goes to zero. We have also studied the temperature dependence of the conductance at the decoupled and Griffiths S-matrices.