We study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the respective Boltzmann weights and found that they possess a universal structure. This allows us to classify the integrable manifolds in four different families reproducing three known models besides uncovering a novel nineteen vertex model in a unified way. The introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic. The diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed. We point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin-1 chains.

http://arxiv.org/abs/1010.1274
Mathematical Physics (math-ph)