Free Fermion Conditions and the Symmetries of Integrability

We show that free-fermion conditions on vertex models amount to degeneration conditions for the inversion relations and the symmetries of the Yang-Baxter equations. We also show how solutions of the tetrahedron equations enjoy a similar property, and analyze the phenomenon.

TAXONOMY OF THE SIXTEEN-VERTEX MODELS

A classification of the subcases of the sixteen-vertex model compatible with the infinite symmetry group generated by the inversion relations of the model is performed. The elliptic parameterization of these models is recalled, emphasizing the subvarieties of the parameter space for which this parameterization degenerates into a rational one. This situation corresponds to the vanishing of some discriminant and is deeply related to the critical and disorder manifolds for these models. We concentrate on subcases of the sixteen-vertex model for which factorizations of this discriminant occur, allowing further exact calculations.

INVESTIGATION OF EXCITATION SPECTRA OF EXACTLY SOLVABLE MODELS USING INVERSION RELATIONS

We study the excitations of two-dimensional classical vertex models and related one-dimensional quantum Hamiltonians. The main mathematical means we use are functional equations which are set up and solved for the excitation functions. The method is first applied to the eight-vertex model as a prototype and compared with the traditional solution procedure of the Bethe Ansatz. In the last part we investigate some q-state models, notably the biquadratic spin-1 quantum antiferromagnet and its generalizations, which recently attracted interest due to Haldane’s conjecture.