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We present a mapping of correlated multi-impurity Anderson models to acluster model coupled to a number of effective conduction bands capturing itsessential low-energy physics. The major ingredient is the complexsingle-particle self energy matrix of the uncorrelated problem that encodes theinfluence to the host conduction band onto the dynamics of a set of correlatedorbitals. While the real part of the self-energy matrix generates an effectivehopping between the cluster orbitals, the imaginary part determines thecoupling to the effective conduction bands in the mapped model. The rank of theimaginary part determines the number of independent screening channels of theproblem, and allows the replacement of the phenomenological exhaustioncriterion by a rigorous mathematical statement. This rank provides adistinction between multi-impurity models of first kind and of second kind. Forthe latter, there are insufficient screening channels available, so that asinglet ground state must be driven by the inter-cluster spin correlations.This classification provides a fundamental answer to the question of whyferromagnetic correlations between local moments are irrelevant for the spincompensated ground state in dilute impurity models, whereas they compete withthe Kondo-scale in dense impurity arrays, without evoking a spin density wave.The low-temperature physics of three examples taken from the literature arededuced from the analytic structure of the mapped model, demonstrating thepotential power of this approach. NRG calculations are presented for up to fivesite cluster. We investigate the appearance of frustration induced non-Fermiliquid fixed points in the trimer, and demonstrate the existence of severalcritical points of KT type at which ferromagnetic correlations suppress thescreening of an additional effective spin-$1/2$ degree of freedom.

Strongly correlated multi-impurity models: The crossover from a single-impurity problem to lattice models