Kac–Ward Formula and Its Extension to Order–Disorder Correlators Through a Graph Zeta Function

A streamlined derivation of the Kac–Ward formula for the planar Ising model’s partition function is presented and applied in relating the kernel of the Kac–Ward matrices’ inverse with the correlation functions of the Ising model’s order–disorder correlation functions. A shortcut for both is facilitated by the Bowen–Lanford graph zeta function relation. The Kac–Ward relation is also extended here to produce a family of non planar interactions on Z² for which the partition function and the order–disorder correlators are solvable at special values of the coupling parameters/temperature.