Multi-impurity Anderson model for quantum dots coupled in parallel

TL;DR

This paper models multiple quantum dots in parallel as an N-impurity Anderson system, revealing a ferromagnetic RKKY interaction, a maximal spin ground state, and a robust N/2 Kondo effect, with implications for quantum phase transitions.

Contribution

It introduces an analytical and numerical study of the N-impurity Anderson model, showing the emergence of a stable N/2 Kondo ground state and its robustness against perturbations.

Findings

The RKKY interaction is ferromagnetic and locks the dots into a maximal spin state.

At low temperatures, the system exhibits an N/2 Kondo effect.

The Kondo temperature is independent of the number of impurities N.

Abstract

The system of several (N) quantum dots coupled in parallel to the same single-mode conduction channel can be modelled 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/2-1/2 due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the N-impuriy 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.