Geometrical aspects of the Z-invariant Ising model

TL;DR

This paper explores the geometric interpretation of the Z-invariant Ising model through isoradial embeddings, linking lattice geometry with model criticality and coupling constants.

Contribution

It establishes a connection between isoradial embeddings and the Z-invariant Ising model, providing a geometric framework for understanding its critical points.

Findings

Isoradial embeddings characterize Z-invariant Ising models.

Topological constraints relate to the existence of isoradial embeddings.

Critical points correspond to models on isoradial embeddings.

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.