Exact solutions for Ising model odd-number correlations on the honeycomb and triangular lattices

Investigating honeycomb and triangular simple spin-12 Ising model ferromagnets, exact algebraic systems of linear identities are developed containing the spontaneous magnetization and other odd-number correlations where the coefficients depend upon the interaction parameters. Aided by star-triangle-type relationships and making supplemental use of only the spontaneous magnetization from the literature, it is found that these identity systems are exactly solvable for the odd-number multisite correlations. The method therefore satisfies closure (non-hierarchal) and linear independence requirements, is relatively transparent, and approximately eighty odd- number localized correlations are obtained containing up to and including nine (seven) sites for the honeycomb (triangular) lattice, and a simple prescription is given and demonstrated for extracting their critical amplitudes. The results also offer examples of correlation degeneracies (both essential and accidental types) and reveal the existence of linear correlation identities which do not depend explicitly upon the interaction parameters.