Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function

Abstract

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 ${\mathbb{Z}}^2$ for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.

An extension of the Kac-Ward determinantal formula beyond planarity was added (Section 5). To appear in Journal of Statistical Physics

References in corpus (3)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Revisiting the combinatorics of the 2D Ising model
• Pfaffian Correlation Functions of Planar Dimer Covers

Cited by in corpus (2)

• Emergent Planarity in two-dimensional Ising Models with finite-range Interactions
• Kac-Ward solution of the 2D classical and 1D quantum Ising models