Narrow Results

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (N_α,N_β) model. This model consists of two scalar Potts models (of N_α and N_β states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If N_α,N_β≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for N_α=N_β; whereas if N_α≠N_β=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

A Cremona transformation X=f(x, y), Y=g(x, y) is a rational mapping (meaning that f and g are ratios of polynomials) with rational inverse x=F(X, Y), y=G(X, Y). Discrete dynamical systems defined by such transformations are well studied. They include symmetries of the Yang-Baxter equations and their generalizations. In this paper we comment on two types of dynamical systems based on Cremona transformations. The first is the P1 case of Bellon et al. which pertains to the inversion relation for the matrix of Boltzmann weights of the 4-state chiral Potts model. The resulting dynamical system decouples completely to one in a single variable. The sub case z=x corresponds to the symmetric Ashkin-Teller model. We solve this case explicitly giving orbits as closed formulas in the number n of iterations. The second type of system treated is an extension from the famous example due to McMillan of invariant curves of area preserving maps in two dimensions to the case of invariant curves and surfaces of three dimensional Cremona maps that preserve volume. The trace map of the renormalization of transmission through a Fibonacci chain, first introduced by Kohmoto, Kadanoff and Tang, is considered as an example of such a system.

By using a Monte Carlo simulation, we have studied the effects on the multilayer transition and magnetic properties of the crystal field $D∕J_2$, the four-spin coupling $J_4∕J_2$ and RKKY interaction in the spin-1 Ashkin–Teller (AT) model of a system composed of two magnetic multi-layer materials, of various thicknesses, separated by a nonmagnetic spacer of thickness $M$. By varying the strength of $J_4∕J_2$, we have obtained a new partially ordered phase $〈\sigma S〉$, for different values of $D∕J_2$. We have established a rich phase diagram with first and second-order phase transitions with tricritical points. It is shown that the transition temperature of both magnetic blocks depends strongly on the thickness of the magnetic layers $N$, as well as $J_4∕J_2$ and $D∕J_2$. There exists a critical thickness of the magnetic layers $N_L$ above which the multilayer blocks undergo a transition separately at different temperatures, for a high value of $J_4∕J_2$ and $D∕J_2$. Besides, the critical exponents are computed.

Two-dimensional toy models display, in a gentler setting, many salient aspects of Quantum Field Theory. Here I discuss a concrete two-dimensional case, the Thirring model, which illustrates several important concepts of this theory: the anomalous dimension of the fields; the exact solvability; the anomalies of the Ward-Takahashi identities. Besides, I give a glimpse of the decisive role that this model plays in the study of an apparently unrelated topic: correlation critical exponents of two dimensional lattice systems of Statistical Mechanics.