Zeros of planar Ising models via flat SU(2) connections

TL;DR

This paper proves a geometric formula for certain complex zeros of the Ising model's partition function on planar graphs, linking these zeros to flat SU(2) connections and graph immersions in three-dimensional space.

Contribution

It rigorously establishes a geometric interpretation of the zeros using SU(2) connections, connecting graph immersions, flatness, and the Kac-Ward matrix.

Findings

Zeros depend on graph immersion geometry

Null eigenvector construction confirms the formula

SU(2) connection flatness relates to eigenvector existence

Abstract

Livine and Bonzom recently proposed a geometric formula for a certain set of complex zeros of the partition function of the Ising model defined on planar graphs. Remarkably, the zeros depend locally on the geometry of an immersion of the graph in the three dimensional Euclidean space (different immersions give rise to different zeros). When restricted to the flat case, the weights become the critical weights on circle patterns. We rigorously prove the formula by geometrically constructing a null eigenvector of the Kac-Ward matrix whose determinant is the squared partition function. The main ingredient of the proof is the realisation that the associated Kac-Ward transition matrix gives rise to an SU(2) connection on the graph, creating a direct link with rotations in three dimensions. The existence of a null eigenvector turns out to be equivalent to this connection being flat.