Quantum impurity in a Luttinger liquid: Exact solution of the Kane-Fisher model

TL;DR

This paper provides an exact solution to the Kane-Fisher model of a quantum impurity in a Luttinger liquid, revealing detailed spectral and thermodynamic properties across all interaction regimes.

Contribution

It introduces an exact Bethe Ansatz solution for the Kane-Fisher and Weak-Tunnelling models, incorporating all scattering processes and establishing a generalized Yang-Baxter relation.

Findings

Derived the spectrum for all coupling regimes.

Calculated the impurity free energy.

Analyzed low energy behavior for different interactions.

Abstract

A Luttinger Liquid coupled to a quantum impurity describes a large number of physical systems. The Hamiltonian consists of left- and right-moving fermions interacting among themselves via a density-density coupling and scattering off a localised transmitting and reflecting impurity. We solve exactly the Hamiltonian by means of an incoming-outgoing scattering Bethe basis which properly incorporates all scattering processes. A related model, the Weak-Tunnelling model, wherein the impurity is replaced by a tunnel junction, is solved by the same method. The consistency of the construction is established through a generalised Yang-Baxter relation. Periodic boundary conditions are imposed and the resulting Bethe Ansatz equations are derived by means of the Off Diagonal Bethe Ansatz approach. We derive the spectrum of the model for all coupling constant regimes and calculate the impurity free energy. We discuss the low energy behaviour of the systems for both repulsive and attractive interactions.