Modified tetrahedron equation in a 3-dimensional Ising model and Hopfield neural network on a triangular lattice

Integrability in statistical physics models is usually expressed in the fact that the partition function can be represented via a transfer matrix included in a "large" commutative family. The latter property for two-dimensional models is traditionally accompanied by a vertex model structure with a weight matrix satisfying the Yang-Baxter equation. The report will focus on the generalization of this idea to a large dimension, in particular, I will consider the three-dimensional Ising model, as well as the Hopfield neural network model on a 2-dimensional triangular lattice in the retrieval phase. It turns out that both of these models have a vertex representation, with a weight matrix satisfying the modified tetrahedron equation. In both cases, the hypercube combinatorics is essentially used to construct the weight matrix.
The report is based on the works:
http://lanl.arxiv.org/abs/1805.04138
http://lanl.arxiv.org/abs/1806.06680