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arXiv:1005.3863

A renormalization group study of the three-color Ashkin-Teller model on a Wheatstone hierarchical lattice

R. Teodoro, Claudionor Gomes Bezerra, Ananias M. Mariz, F.A. da Costa

Published May 21, 2010• 0 citations

Abstract

We have investigated the three-color Ashkin-Teller model (3AT), on the Wheatstone bridge hierarchical lattice, by means of a Migdal-Kadanoff renormalization group approach. We have obtained the exact recursion relations for the renormalized couplings, which have been used to investigate the phase diagram and to study the corresponding critical points. The phase diagram, represented in terms of the dual transmissivity vector, presents four magnetic phases and nine critical points. We have also numerically calculated the correlation length ($ν_T$) and crossover ($ϕ$) critical exponents, which show that seven of the critical points are in the Potts model universality class ($q=2$, 4 e 8). The remaining critical points are in a universality class which may belong to the Baxter's line. Our results are exact on the hierarchical lattice used in the present work and the phase diagram can be considered as an approximation to more realistic Bravais lattices.

DOI: 10.48550/arxiv.1005.3863

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We have investigated the three-color Ashkin-Teller model (3AT), on the Wheatstone bridge hierarchical lattice, by means of a Migdal-Kadanoff renormalization group approach. We have obtained the exact recursion relations for the renormalized couplings, which have been used to investigate the phase diagram and to study the corresponding critical points. The phase diagram, represented in terms of the dual transmissivity vector, presents four magnetic phases and nine critical points. We have also numerically calculated the correlation length (

ν_T

) and crossover (

ϕ

) critical exponents, which show that seven of the critical points are in the Potts model universality class (

q=2

, 4 e 8). The remaining critical points are in a universality class which may belong to the Baxter's line. Our results are exact on the hierarchical lattice used in the present work and the phase diagram can be considered as an approximation to more realistic Bravais lattices.

The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.

Using the real-space renormalization-group transformation, we study the phase transitions of the Ashkin-Teller model including the antiferromagnetic interactions on a type of diamond hierarchical lattices, of which the number of bonds per branch of the generator is odd. The isotropic Ashkin-Teller model and the anisotropic one are, respectively, investigated. We find that the phase diagram, for the isotropic Ashkin-Teller model, consists of five phases, two of which are associated with the partially antiferromagnetic ordering of the system, while the phase diagram, for the anisotropic Ashkin-Teller model, contains 11 phases, six of which are related to the partially antiferromagnetic ordering of the system. The correlation length critical exponents and the crossover exponents are also calculated.

The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (

n=1

) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.

The critical behavior of the

s

-state Ashkin-Teller-Potts model on a square lattice is studied for

s=3, 4, \mathrm{and} 5

, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the

s=3

and

s=4

models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For

s=5

, however, the transformation fails to detect the first-order transition that is known to occur.

Within a real space renormalisation group approach, the authors study the phase diagram and the universality classes of the three-state chiral clock model in a self-dual planar hierarchical lattice. They find that the chiral field Delta is relevant at the pure Potts critical point with a crossover exponent phi approximately=0.32. The critical line which separates the ferromagnetic phase from the non-ferromagnetic ones is characterised by a multicritical (Lifshitz) point located at (K, Delta ) approximately= (1.452, 0.382).

We investigate the zero-temperature quantum phase transitions of the disordered three-color quantum Ashkin-Teller spin chain by means of large-scale Monte Carlo simulations. We find that the first-order phase transitions of the clean system are rounded by the quenched disorder. For weak intercolor coupling, the resulting emergent quantum critical point between the paramagnetic phase and the magnetically ordered Baxter phase is of infinite-randomness type and belongs to the universality class of the random transverse-field Ising model, as predicted by recent strong-disorder renormalization group calculations. We also find evidence for unconventional critical behavior in the case of strong intercolor coupling, even though an unequivocal determination of the universality class is beyond our numerical capabilities. We compare our results to earlier simulations, and we discuss implications for the classification of phase transitions in the presence of disorder.