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arXiv:cond-mat/9508110 · Physical review. B, Condensed matter

Exact finite-size spectrum for the multi-channel Kondo model and Kac-Moody fusion rules

Satoshi Fujimoto, Norio Kawakami

Published Nov 1, 1995• 4 citations
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Abstract

The finite-size spectrum for the multi-channel Kondo model is derived analytically from the exact solution, by mapping the nontrivial ${\mathit{Z}}_{\mathit{n}}$ part of the Kondo scattering into that for the restricted solid-on-solid model coupled with the impurity. The analysis is performed for the case of n-2S=1, where n is the number of channels and S is the impurity spin. The result obtained is in accordance with the Kac-Moody fusion hypothesis proposed by Affleck and Ludwig.

DOI
10.1103/physrevb.52.r13102

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The finite-size spectrum for the multi-channel Kondo model is derived analytically from the exact solution, by mapping the nontrivial Zn{\mathit{Z}}_{\mathit{n}}Zn​ part of the Kondo scattering into that for the restricted solid-on-solid model coupled with the impurity. The analysis is performed for the case of n-2S=1, where n is the number of channels and S is the impurity spin. The result obtained is in accordance with the Kac-Moody fusion hypothesis proposed by Affleck and Ludwig.
The multichannel Kondo model supports effective anyons on the partially screened impurity, as suggested by its fractional impurity entropy. It was recently demonstrated for the multi-impurity chiral Kondo model, that scattering of an electron through the impurities depends on the anyon's total fusion channel. Here we study the correlation between impurity spins. We argue, based on a combination of conformal field theory, a perturbative limit with a large number of channels kkk, and the exactly solvable two-channel case, that the interimpurity spin correlation probes the anyon fusion of the pair of correlated impurities. This may allow, using measurement-only topological quantum computing protocols, to braid the multichannel Kondo anyons via consecutive measurements.
We solve the two-impurity, two-channel Kondo model using a combination of conformal invariance and bosonization techniques. In the odd-even symmetric case, the RKKY interaction is exactly marginal, resulting in a line of non-Fermi-liquid fixed points. Explicit formulas are given for the critical exponents and for the finite-size spectrum, which depend continuously on a single parameter. The marginal line spans a range of values of the RKKY coupling III which goes from the infinitely strong ferromagnetic point $I\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{\infty}$ to a finite antiferromagnetic critical value Imin<0{I}_{\mathrm{min}}<0Imin​<0 beyond which a Fermi liquid is recovered. When the odd-even symmetry is broken, the marginal line is unstable for ferromagnetic III, while for antiferromagnetic III it extends into a manifold of fixed points.
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.
We review analytical and numerical results derived from the Bethe ansatz solution of the n-channel Kondo model of arbitrary spin S as a function of temperature, external field, impurity spin S and the number of channels n. Three situations have to be distinguished: (i) If n = 2S the conduction electrons exactly compensate the impurity spin into a singlet at low temperatures, (ii) if n < 2S the impurity spin is only partially compensated (undercompensated), and (iii) if n > 2S the impurity spin is said to be overcompensated giving rise to critical behaviour. The results are discussed in the context of magnetic impurities, e.g. Fe, Cr and Tm in simple metals, the quadrupolar Kondo effect, an impurity spin embedded in the Takhtajan-Babujian Heisenberg model and electron assisted-tunnelling of an atom in a double-well potential.