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A microscopic derivation of the classical Generalized Constant Coupling (GCC)model for geometrically frustrated magnets is presented. Starting from theclassical Heisenberg Hamiltonian, the partition function for clusters with p =2, 3, 4 ,...spins in the presence of the inhomogeneous symmetry breaking fields(SBF) created by spins outside the unit is calculated. The effective fieldscharacterizing the interaction between units naturally arise as averages overthe SBF. These effective fields are fixed by a self-consistency condition. Inthe paramagnetic regime, we recover all the results previously obtained in amore phenomenological way, which were shown to be in excellent agreement withMonte Carlo calculations for these lattices. In the absence of applied magneticfield, it is found that, for antiferromagnetic interactions, the equilibriumconfiguration is a non-collinear configuration in which the total magnetizationof the unit is zero, and the condition under which such an ordered state occursis also obtained from the calculation.  However, frustration inhibits the formation of such an state, and the systemremains paramagnetic down to 0 K, if only nearest neighbor interactions aretaken into account, for all the systems considered. Furthermore, we also studythe effect of next nearest neighbor interactions (NNN) and site dilution bynon-magnetic impurities for the pyrochlore lattice. It is found that NNNinteractions can stabilize a non-collinear ordered state, or ferromagnetic one,depending on the relation between NN and NNN interactions, and the phasediagram is calculated. However, site dilution is not enough by itself to formsuch an ordered state.Comment: 26 pages, 9 figures, RevTeX

physical review. b, condensed matter

Classical generalized constant-coupling method for geometrically frustrated magnets: Microscopic formulation and effect of perturbations beyond nearest-neighbor interactions