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Non-perturbative $J_{pd}$ model and ferromagnetism in dilute magnets

We calculate magnetic couplings in the $J_{pd}$ model for dilute magnets, in order both to identify the relevant parameters which control ferromagnetism and also to bridge the gap between first principle calculations and model approaches. The magnetic exchange interactions are calculated non-perturbatively and disorder in the configuration of impurities is treated exacly, allowing us to test the validity of effective medium theories. Results differ qualitatively from those of weak coupling. In contrast to mean field theory, increasing $J_{pd}$ may not favor high Curie temperatures: $T_C$ scales primarily with the bandwidth. High temperature ferromagnetism at small dilutions is associated with resonant structure in the p-band. Comparison to diluted magnetic semiconductors indicate that Ga(Mn)As has such a resonant structure and thus this material is already close to optimality.