On nonintegrability of three-dimensional Ising model

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 (L-T) expansion contains quartic terms, which do not allow explicit computation of the integral. We wanted to check — apparently not explored — the possibility that using the high-temperature (H-T) expansion would result in action with only bilinear terms (in two dimensions, L-T and H-T expansions are equivalent, but in three dimensions, they differ from each other). It turned out, however, that polygons obtained by Grassmann 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 Grassmann integral over bilinear action.