Eur. Phys. J. B 53, 85-90 (2006)
https://doi.org/10.1140/epjb/e2006-00336-1

Geometrical aspects of the Z-invariant Ising model

Spinoza Institute, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands

Corresponding author: ^a R.A.Costa-Santos@phys.uu.nl

Received: 18 May 2004
Revised: 21 August 2005
Published online: 22 August 2006

Abstract

We discuss a geometrical interpretation of the Z-invariant Ising model in terms of isoradial embeddings of planar lattices. The Z-invariant Ising model can be defined on an arbitrary planar lattice if and only if certain paths on the lattice edges do not intersect each other more than once or self-intersect. This topological constraint is equivalent to the existence of isoradial embeddings of the lattice. Such embeddings are characterized by angles which can be related to the model coupling constants in the spirit of Baxter's geometrical solution. The Ising model on isoradial embeddings studied recently by several authors in the context of discrete holomorphy corresponds to the critical point of this particular Z-invariant Ising model.