Simple Bosonization Solution of the 2-channel Kondo Model: I. Analytical Calculation of Finite-Size Crossover Spectrum

TL;DR

This paper provides an exact analytical solution to the anisotropic 2-channel Kondo model at its Toulouse point, clarifying the finite-size crossover spectrum and connecting various renormalization group methods.

Contribution

It generalizes the bosonization-refermionization approach to finite systems, explicitly constructs boson fields and Klein factors, and offers a detailed analytic description of the crossover to the non-Fermi-liquid fixed point.

Findings

Explicit construction of boson and Klein factors in original fermion operators

Analytic description of the finite-size crossover spectrum

Connection between different renormalization group schemes

Abstract

We present in detail a simple, exact solution of the anisotropic 2-channel Kondo (2CK) model at its Toulouse point. We reduce the model to a quadratic resonant-level model by generalizing the bosonization-refermionization approach of Emery and Kivelson to finite system size, but improve their method in two ways: firstly, we construct all boson fields and Klein factors explicitly in terms of the model's original fermion operators $c_{k \sigma j}$, and secondly we clarify explicitly how the Klein factors needed when refermionizing act on the original Fock space. This enables us to explicitly follow the adiabatic evolution of the 2CK model's free-fermion states to its exact eigenstates, found by simply diagonalizing the resonant-level model for arbitrary magnetic fields and spin-flip coupling strengths. In this way we obtain an {\em analytic} description of the cross-over from the free to the non-Fermi-liquid fixed point. At the latter, it is remarkably simple to recover the conformal field theory results for the finite-size spectrum (implying a direct proof of Affleck and Ludwig's fusion hypothesis). By analyzing the finite-size spectrum, we directly obtain the operator content of the 2CK fixed point and the dimension of various relevant and irrelevant perturbations. Our method can easily be generalized to include various symmetry-breaking perturbations. Furthermore it establishes instructive connections between different renormalization group schemes such as poor man's scaling, Anderson-Yuval type scaling, the numerical renormalization group and finite-size scaling.