Geometric Formula for 2d Ising Zeros: Examples & Numerics

TL;DR

This paper tests a geometric formula predicting the zeros of the 2d Ising model's partition function using analytical and numerical methods on various triangulations, revealing its limitations on non-trivial topologies.

Contribution

It provides an analytical verification and extensive numerical validation of the geometric formula for 2d Ising zeros on planar triangulations, and identifies its failure on toroidal topologies.

Findings

The formula accurately predicts zeros on planar triangulations.

The formula does not extend to toroidal topologies without modifications.

Numerical methods confirm the geometric interpretation of Ising zeros.

Abstract

A geometric formula for the zeros of the partition function of the inhomogeneous 2d Ising model was recently proposed in terms of the angles of 2d triangulations embedded in the flat 3d space. Here we proceed to an analytical check of this formula on the cubic graph, dual to a double pyramid, and provide a thorough numerical check by generating random 2d planar triangulations. Our method is to generate Delaunay triangulations of the 2-sphere then performing random local rescalings. For every 2d triangulations, we compute the corresponding Ising couplings from the triangle angles and the dihedral angles, and check directly that the Ising partition function vanishes for these couplings (and grows in modulus in their neighborhood). In particular, we lift an ambiguity of the original formula on the sign of the dihedral angles and establish a convention in terms of convexity/concavity. Finally, we extend our numerical analysis to 2d toroidal triangulations and show that the geometric formula does not work and will need to be generalized, as originally expected, in order to accommodate for non-trivial topologies.