Universal relations for nonsolvable statistical models.

We present the first rigorous derivation of a number of universal relations for a class of models with continuously varying indices (among which are interacting planar Ising models, quantum spin chains and 1D Fermi systems), for which an exact solution is not known, except in a few special cases. Most of these formulas were conjectured by Luther and Peschel, Kadanoff, and Haldane, but only checked in the special solvable models; one of them, related to the anisotropic Ashkin-Teller model, is novel.

Anomalous critical exponents in the anisotropic Ashkin-Teller model.

We perform a rigorous computation of the specific heat of the Ashkin-Teller model in the case of small interaction and we explain how the universality-nonuniversality crossover is realized when the isotropic limit is reached. We prove that, even in the region where universality for the specific heat holds, anomalous critical exponents appear: for instance, we predict the existence of a previously unknown anomalous exponent, continuously varying with the strength of the interaction, describing how the difference between the critical temperatures rescales with the anisotropy parameter.