Exact solutions for Ising-model even-number correlations on planar lattices

A systematic and unifying method is developed and demonstrated for obtaining exact solutions of n-site (n even integer) Ising correlations on various planar lattices. The scheme, which is exceedingly more simple than using solely traditional Pfaffian techniques, embodies five mapping theorems in alliance with algebraic correlation identities. In the theoretical framework, the triangular Ising model plays an overarching role. In particular, considering a select 7-site cluster of the triangular Ising model, the knowledge of all its 11 even-number correlations defined upon this cluster (where only four of the correlations need to be actually calculated by Pfaffian procedures) is shown to be sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagomé Ising model even-number correlations upon their correspondingly select 10-site, 19-site, and 9-site clusters, respectively. The relative ease and direct applicability of the present approach are highlighted not only by the resulting large numbers of n-site (n even-integer) correlation solutions (e.g., approximately 85 and 50 for the honeycomb and kagomé Ising models, respectively) and their large $n_{\max }$ values ($n_{\max =8\mathrm{}}$,10,18 for the kagomé, honeycomb, and decorated-honeycomb Ising models, respectively) but also by the realization that the exact solutions for Ising multisite correlations upon the kagomé lattice (one of the four regular lattices in two dimensions) are apparently the first to explicitly appear in the literature beyond its nearest-neighbor pair correlation (energy).