On the non-integrability of three dimensional Ising model

TL;DR

This paper demonstrates that the three-dimensional Ising model's partition function cannot be expressed as a Grassmann integral with only bilinear terms, indicating non-integrability in this context.

Contribution

It shows that unlike the two-dimensional case, the 3D Ising model's Grassmann representation cannot be simplified to bilinear form, highlighting its non-integrability.

Findings

Polygons in Grassmann integration do not have fixed signs in 3D

High-temperature expansion does not yield a bilinear Grassmann action

Partition function cannot be expressed as a bilinear Grassmann integral in 3D

Abstract

It is well known that the partition function of two-dimensional Ising model can be expressed as a Grassmann integral over the action bilinear in Grassmann variables. The key aspect of the proof of this equivalence is to show that all polygons, appearing in Grassmann integration, enter with fixed sign. For three-dimensional model, the partition function can also be expressed by Grassmann integral. However, the action resulting from low-temperature expansion contains quartic terms, which does not allow explicit computation of the integral. We wanted to check - apparently not explored - the possibility that using the high-temperature expansion would result in action with only bilinear terms. (in two dimensions, low-T and high-T expansions are equivalent, but in three dimensions, they differ.) It turned out, however that polygons obtained by Grassman integration are not of fixed sign for any ordering of Grassmann variables on sites. This way, it is not possible to express the partition function of three-dimensional Ising model as a Grassman integral over bilinear action.