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

摘要

We present in detail a simple, exact solution of the anisotropic 2-channelKondo (2CK) model at its Toulouse point. We reduce the model to a quadraticresonant-level model by generalizing the bosonization-refermionization approachof Emery and Kivelson to finite system size, but improve their method in twoways: firstly, we construct all boson fields and Klein factors explicitly interms of the model's original fermion operators $c_{k \sigma j}$, and secondlywe clarify explicitly how the Klein factors needed when refermionizing act onthe original Fock space. This enables us to explicitly follow the adiabaticevolution of the 2CK model's free-fermion states to its exact eigenstates,found by simply diagonalizing the resonant-level model for arbitrary magneticfields and spin-flip coupling strengths. In this way we obtain an {\emanalytic} description of the cross-over from the free to the non-Fermi-liquidfixed point. At the latter, it is remarkably simple to recover the conformalfield theory results for the finite-size spectrum (implying a direct proof ofAffleck and Ludwig's fusion hypothesis). By analyzing the finite-size spectrum,we directly obtain the operator content of the 2CK fixed point and thedimension of various relevant and irrelevant perturbations. Our method caneasily be generalized to include various symmetry-breaking perturbations.Furthermore it establishes instructive connections between differentrenormalization group schemes such as poor man's scaling, Anderson-Yuval typescaling, the numerical renormalization group and finite-size scaling.