The Quantum Statistical Free Energy Minimum Principle for Multi-Lattice Mean Field Theories

The quantum statistical frame for infinite multi-lattice spin systems is introduced. The thermodynamic functionals specific internal energy, entropy and free energy are shown to exist on the set of permutation invariant states for polynomial mean field interactions by direct estimation methods. Their dependence on the relative sizes of the sublattice systems is made explicit. The set of homogeneous minimal free energy states is shown to be a Bauer simplex which contains all limiting Gibbs states. For the extremal minimal free energy states the self-consistency equations are derived.