The system of several
(N) quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel
N-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the
N-impurity
S=1∕2 Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin
S=N∕2 ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the
N-impurity Anderson model to the exact Bethe ansatz results of a
S=N∕2 SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective
S=N∕2 Kondo model. Moreover, the Kondo temperature is independent of the number of impurities
N. We demonstrate the robustness of the spin
N∕2 ground state as well as of the associated
S=N∕2 Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.
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.
We propose an integrable model of the spin-1/2 Heisenberg chain coupled to two impurity moments. With the open boundary conditions at the impurity sites, the model can be exactly solved for arbitrary impurity spin and arbitrary exchange constants between the bulk and the impurities. The absence of redundant terms in the Hamiltonian makes the model very reasonable. The Hamiltonian is diagonalized via algebraic Bethe ansatz. It is found that the impurity spins can only be screened (partially for
S>1/2) for antiferromagnetic coupling between the impurity and the bulk. Otherwise the impurity spins cannot be screened. The residual entropy of the ground state and the Kondo temperature are also derived explicitly based on the Bethe ansatz and the local Fermi-liquid theory.
A complete Bethe ansatz solution of the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)$ Coqblin-Schrieffer model and a detailed analysis of some physical applications of the model are given. As in the usual multichannel Kondo model, a variety of Fermi-liquid and non-Fermi-liquid (NFL) fixed points is found, whose nature depends on the impurity representation \ensuremath{\mu}. For $\ensuremath{\mu}=f,$ we find a Fermi-liquid fixed point, with the impurity spin completely screened. For $f>\ensuremath{\mu},$ the impurity is overscreened and the model has NFL properties. The form the NFL behavior takes depends on the
N and
f: for
N< f, the specific heat and the susceptibility are dominated by the NFL contributions; for
N>f the leading contributions are Fermi-liquid-like, and the NFL behavior can be seen only to subleading order; and for
N=f the behavior is marginal. We also analyze the possibility of physical realizations. We show by a detailed renormalization-group and
1/f analysis that the tunneling
N-state problem can be mapped into the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(f)$ exchange model, and discuss the subtle differences between the two models. As another physical realization we suggest a double quantum dot structure that can be described by means of an $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ model if the parameters of the dots are tuned appropriately.
A single multichannel quantum impurity model with pure hybridizations between the localized and orbital degenerate conducting electrons is shown to exhibit several interesting impurity properties resembling that of the multichannel Anderson model. Particularly for two special kinds of hybridization parameters the model can be solved exactly by the Bethe ansatz method. In these integrable cases, the competition between the direct and exchange correlated hybridizations can produces either a Fermi-liquid or non-Fermi-liquid behavior at low temperatures.