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Physical review. B, Solid state

Investigation of the Potts Model Using Renormalization-Group Techniques

Geoffrey R. Golner

Published Oct 1, 1973• 112 citations
Publisher

Abstract

We report the numerical study, using Wilson's $\ensuremath{\eta}=0$ approximate renormalization-group recursion formula, of a continuous-spin generalization of the three-component Potts model. Previous numerical studies, based on series expansions of the partition function, have indicated strongly that the model has a phase transition of second order. This is in conflict with Landau theory which predicts that the transition be of first order. We find, in agreement with Landau theory, that our model exhibits a first-order phase transition having finite zero-field spontaneous magnetization at the transition point.

DOI
10.1103/physrevb.8.3419

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A generalization of the Niemeijer-van Leeuwen renormalization-group transformation treating disordered cells of spins as vacancies is introduced and applied to the two-dimensional qqq-state Potts model. Finite-lattice approximations yield the changeover from continuous to first-order phase transitions predicted by Baxter but not observed in previous renormalization-group calculations. Exact results are conjectured for the tricritical exponents of dilute Potts models.
The Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays, for Jnnn/Jnn = 0.1, three phase transitions in different universality classes as the magnetic field is increased. We have studied this model using Monte Carlo and renormalization group techniques. The transition from the paramagnetic to the 2 × 1 phase (universality class of the Heisenberg model with cubic anisotropy) is found to be first order; the transition from the paramagnetic phase to the [Formula: see text] phase (universality class of the three state Potts model) is continuous; and the transition from the paramagnetic to the 2 × 2 phase (universality class of the four state Potts model) is found to change from first order to continuous as the field is increased. We have mapped out the phase diagram and determined the critical exponents for the continuous transitions. A novel technique, using a Landau-like free energy functional determined from Monte Carlo calculations, to distinguish between first order and continuous transitions, is described.
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