Fases e criticalidade no modelo ashkin - teller de três cores - Piolho, Francisco de Assis Pereira (2007)
The usual Ashkin-Teller (AT) model is obtained as a superposition of two Ising models coupled through a four-spin interaction term. In two dimension the AT model displays a line of fixed points along which the exponents vary continuously. On this line the model becomes soluble via a mapping onto the Baxter model. Such richness of multicritical behavior led Grest and Widom to introduce the N-color Ashkin-Teller model (N-AT). Those authors made an extensive analysis of the model thus introduced both in the isotropic as well as in the anisotropic cases by several analytical and computational methods. In the present work we define a more general version of the 3-color Ashkin-Teller model by introducing a 6-spin interaction term. We investigate the corresponding symmetry structure presented by our model in conjunction with an analysis of possible phase diagrams obtained by real space renormalization group techniques. The phase diagram are obtained at finite temperature in the region where the ferromagnetic behavior is predominant. Through the use of the transmissivities concepts we obtain the recursion relations in some periodical as well as aperiodic hierarchical lattices. In a first analysis we initially consider the two-color Ashkin-Teller model in order to obtain some results with could be used as a guide to our main purpose. In the anisotropic case the model was previously studied on the Wheatstone bridge by Claudionor Bezerra in his Master Degree dissertation. By using more appropriated computational resources we obtained isomorphic critical surfaces described in Bezerra's work but not properly identified. Besides, we also analyzed the isotropic version in an aperiodic hierarchical lattice, and we showed how the geometric fluctuations are affected by such aperiodicity and its consequences in the corresponding critical behavior. Those analysis were carried out by the use of appropriated definitions of transmissivities. Finally, we considered the modified 3-AT model with a 6-spin couplings. With the inclusion of such term the model becomes more attractive from the symmetry point of view. For some hierarchical lattices we derived general recursion relations in the anisotropic version of the model (3-AAT), from which case we can obtain the corresponding equations for the isotropic version (3-IAT). The 3-IAT was studied extensively in the whole region where the ferromagnetic couplings are dominant. The fixed points and the respective critical exponents were determined. By analyzing the attraction basins of such fixed points we were able to find the three-parameter phase diagram (temperature £ 4-spin coupling £ 6-spin coupling). We could identify fixed points corresponding to the universality class of Ising and 4- and 8-state Potts model. We also obtained a fixed point which seems to be a sort of reminiscence of a 6-state Potts fixed point as well as a possible indication of the existence of a Baxter line. Some unstable fixed points which do not belong to any aforementioned q-state Potts universality class was also found