The numerical solution of the Bethe ansatz equations of the n-channel Kondo problem is presented. The thermodynamics of the impurity is obtained as a function of temperature, external field, impurity spin S and the number of channels. 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 briefly discussed in the context of magnetic impurities, the quadrupolar Kondo effect, an impurity spin embedded in the Takhtajan-Babujian model and a two-level system interacting with conduction electrons.
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.
The Kondo problem, which describes the interaction of a spin
s magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in
1/s with the renormalization group. We study both the case of an impurity interacting with a single channel
K=1 of fermions in the
s→∞ limit and the case with
K channels in the double-scaling limit
K→∞,
s→∞,
K/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large
s and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime
K>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
We construct and solve numerically the thermodynamic Bethe-ansatz equations for the spin-anisotropic two-channel Kondo model in arbitrary external field h. At high temperatures the specific heat and the susceptibility show power-law dependence. For $\stackrel{\ensuremath{\rightarrow}}{h}0$ and at temperatures below the Kondo temperature
TK a two-channel Kondo effect develops characterized by a Wilson ratio
8/3, and a logarithmic divergence of the susceptibility and the linear specific-heat coefficient. A finite magnetic field,
h>0, drives the system to a Fermi-liquid fixed point with an unusual Wilson ratio which depends sensitively on h.
The low-temperature behavior of a magnetic impurity of spin S interacting with an electron gas via an anisotropic spin exchange is studied via Bethe’s ansatz. The multichannel Kondo model with U(1) invariance is integrable as a function of two continuous (the exchange and the anisotropy) and two discrete parameters, namely the impurity spin S and the number of channels n. As a function of S and n we distinguish: (i) the compensated case with n=2S, (ii) the overcompensated case if n>2S, and (iii) the undercompensated case (n<2S). While in case (i) the ground state is a singlet, the cases (ii), and (iii) yield quantum critical points. The undercompensated one is of a new type with the critical exponents depending on the anisotropy.
We study the low-temperature thermodynamics of a spin-
S magnetic impurity coupled to
m> 2 degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting electrons or it acquires a new leading term governed by a critical exponent ${\ensuremath{\alpha}}_{m}{=(g}_{m}^{\ensuremath{-}1}\ensuremath{-}1)/m,$ where
gm< 1 is a generalized (
m channel) Luttinger-liquid charge parameter measuring the strength of the repulsive electron-electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened
(m=2S).