On the Dimer Solution of Planar Ising Models

Abstract

Derivations of the partition function of the Ising model on a general planar lattice L, which proceed via an associated dimer problem and use Pfaffians, are simplified by constructing a lattice L^Δ (the ``terminal lattice'' derived from an ``expanded lattice'' of L) for which (A) the allowed dimer configurations are in one‐one correspondence with allowed Ising polygon configurations on L, and which (B) is planar if L is planar so that Kasteleyn's theorem may be used directly to construct the appropriate Pfaffian. This is in contrast to previous use of nonplanar associated dimer lattices for which the correspondence is not one‐one, so that is has been necessary to prove a somewhat obscure ``cancellation theorem.''