Z. Phys. B - Condensed Matter 54, 201-206 (1984) Condensed Zeitschrift Matter ftir Physik B Springer-Verlag 1984 Solution of the n-Channel Kondo Problem (Scaling and Integrability) A.M. Tsvelick and P.B. Wiegmann L.D. Landau Institute for Theoretical Physics, USSR Academy of Sciences, Moscow, USSR Received November 14, 1983 The exact solution of the Kondo model for n-flavours of electrons with the spin 1/2 scattered by the S-spin impurity is presented. For n=2S=5 the model describes manganese impurities dissolved in a metal. It is shown that at n>2S the effective exchange coupling approaches a finite fixed point as the energy scale decreases. It means that at n>2S the Gell-Mann-Low function turns to zero in this point and the scaling behaviour of physical quantities is observed. The scaling behaviour, first ob- tained in the 1D quantum field theory, can be analyzed on the basis of the exact solution. In the case n<2S the effective coupling becomes infinitely strong at low energies. 1. Introduction In all known asymptotically free I D quantum field theories the effective interaction rapidly increases with a decrease of the energy scale until the unitary limit is achieved. Such a dramatical role of the in- teraction in the formation of the ground state of the system is primarily due to one-dimensionality and the common lore is that in such systems the fixed point can be either zero or infinite. Recently Nozieres and Blandin [1] have pointed out the first exception to this rule. They have given simple plausible arguments that the fixed point of the exchange Hamiltonian (4), describing the scatter- ing of n-flavours of 1/2-spin electrons by the S-spin impurity (the so-called n-channel Kondo problem), at n>2S corresponds to a finite J*. As a con- sequence, all physical quantities should have the scaling power law at small energy scales. Their pre- diction has been confirmed by the numerical re- normalization group calculations [2]. In this paper we give the Exact Solution of the Nozieres-Blandin Hamiltonian. We show that the scaling can actually exist in the 1D quantum field theory and, moreover, the scaling behaviour can be investigated in the framework of a nontrivial com- pletely integrable system. Below we calculate only the magnetic field depen- dence of the impurity magnetization Mimp(H ). Ac- cording to the general properties of renormaliza- bility, what kind of low field behaviour should be expected? The magnetization satisfies the equation dMimp dlnH -f(z) (1) where the effective coupling z is defined by the Gell- Mann-Low function ~(z) = In I-I/Tx, ; (2) or dlnH ='~'z'; \ dz / If in the case we are interested in, the S-function has a zero at some z = z* fi(z)=fio.(Z-Z* ) at Iz-z*[<l then as a consequence, [3, 4] z-z*c~(H/TtJ ~ at H~0 and (3) l H \:(z*) Mimp(H) o z ~ ] at H~O Below we calculate the impurity magnetization for an arbitrary magnetic field and parameters n and S.