Random matrix theory and classical statistical mechanics: Spin models

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for nonintegrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappear, and the Poisson distribution be recovered on (non Yang-Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion, we tested the generic calculability of the model on these algebraic varieties.