Analytical calculation of the finite-size crossover spectrum of the anisotropic two-channel Kondo model

We present a conceptually simple, analytical calculation of the finite-size crossover spectrum of the anisotropic two-channel Kondo (2CK) model at its Toulouse point. We use Emery and Kivelson’s method, generalized in two ways. First, we construct all boson fields and Klein factors explicitly in terms of the model’s original fermion operators and, secondly, we clarify explicitly how the Klein factors needed when refermionizing act on the original Fock space. This enables us to follow the evolution of the 2CK model’s free-fermion states to its exact eigenstates for arbitrary magnetic fields and spin-flip coupling strengths. We thus obtain an analytic description of the crossover of the finite-size spectrum to the non-Fermi-liquid fixed point, where we recover the conformal field theory results (implying a direct proof of Affleck and Ludwig’s fusion hypothesis). From the finite-size spectrum we extract 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, and to study the crossover to other fixed points produced by these. Furthermore, it establishes instructive connections between different renormalization group schemes. We also apply our method to the single-channel Kondo model.