Zeros of Planar Ising Models via Flat SU(2) Connections

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 Livine and Bonzom (Phys Rev D 111(4):046003, 2025). Remarkably, the zeros depend locally on the geometry of an immersion of the graph in 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 Lis (Commun Math Phys 370(2):507-530, 2019).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 $\text{SU}\left(2\right)$ 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.