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Journal of Physics C Solid State Physics

The square-lattice three-state Potts model with spin anisotropy

Keith De'Bell

Published Apr 20, 1982• 0 citations
Publisher

Abstract

A real-space renormalisation-group calculation is used to obtain a qualitative description of the effects of spin anisotropy on the critical behaviour of the three-state Potts model on a square lattice. It is demonstrated that these effects are essentially the same as those induced by the application of an external field to the isotropic model. Thermodynamic functions are plotted as functions of temperature and the anisotropy parameter.

DOI
10.1088/0022-3719/15/11/017

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A real-space renormalisation-group calculation is used to obtain a qualitative description of the effects of spin anisotropy on the critical behaviour of the three-state Potts model on a square lattice. It is demonstrated that these effects are essentially the same as those induced by the application of an external field to the isotropic model. Thermodynamic functions are plotted as functions of temperature and the anisotropy parameter.
The choice of a convenient self-dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q-state Potts ferromagnet criticality. The exact critical frontier and dimensionality crossover exponent Hf as well as the expected universality behaviour (renormalisation flow sense) are recovered for any linear scaling factor b and all values of q(q<or=4). The b=2 and b=3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den Nijs conjecture. The same calculation is performed, for all values of b for the exponent nu (d=1) associated with the one-dimensional limit and the exact result nu (d=1)=1 is recovered in the limit b to infinity .
We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.