On the bifurcations occuring in the parameter space of the sixteen vertex model: I. Discrete bifurcations

Abstract

The 16-dimensional parameter space of homogeneous sixteen vertex models is scanned for bifurcation points, i.e. points corresponding to models which possess extra symmetries not existing in nearby points. Equivalence classes of models having the same partition function are identified by means of a characteristic “normal” model, represented by a (4 x 4)-diagonal matrix N, and a pair of (2 x 2)-matrices A and B. In this paper the matrix N is assumed to be non-degenerate and the only bifurcations found are those associated with special types of matrices A and B, i.e. matrices whose decomposition in terms of Pauli-matrices corresponds to a vector a ≡ ( a 1, a 2, a 3) or b ≡ ( b 1, b 2, b 3) that is invariant with respect to one or more elements of the cubic symmetry group. The various bifurcation classes of models found include: a) the families of general- and “complementary” eight vertex models, b) discrete sets of doubly- and one-sided cyclic models and c) a number of secondary bifurcation classes within the eight-vertex families, among which is Baxters symmetric eight-vertex model.