Multiple-impurity Anderson model for quantum dots coupled in parallel

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.