On the classification of the homogeneous 16-vertex models on a square lattice: I. Connection with the boson Bogoliubov transformation

Abstract

We consider the classification of all homogeneous 16-vertex models on a square lattice which was proposed by Gaaff and Hijmans. We show that this classification can be translated into (that is, reformulated in terms of) a classification of matrices which closely resembles one already investigated with some success in the literature, viz. the classification connected with the boson Bogoliubov transformation in quantum mechanics. The main difference between our approach and the one of Gaaff and Hijmans who also made a reformulation, is that we can keep our considerations, in conformity with the actual physical problem, within the field of real rather than complex numbers. In the accompanying paper this leads to results with features not present in the analysis over C , which point to a possible connection with a new kind of phase transitions.