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Journal of Physics A Mathematical and General

Variational renormalisation-group approach to the q-state Potts model in two dimensions

Bernard Nienhuis, Eberhard K. Riedel, Michael Schick C

Published Feb 1, 1980• 75 citations
Publisher

Abstract

The two-dimensional q-state Potts model is investigated by means of a Kadanoff lower-bound renormalisation-group transformation that utilises a recent suggestion to identify disordered cells with vacancies. The topology of the phase diagram is obtained, including first- and second-order transitions for q>q c and q<or=q c , respectively, as well as accurate results for critical and tricritical exponents and the critical value q c .

DOI
10.1088/0305-4470/13/2/006

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Journal of Physics A Mathematical and General•Jun 1, 1980•236 citations
The two-dimensional q-state Potts model is investigated by means of a Kadanoff lower-bound renormalisation-group transformation that utilises a recent suggestion to identify disordered cells with vacancies. The topology of the phase diagram is obtained, including first- and second-order transitions for q>q c and q<or=q c , respectively, as well as accurate results for critical and tricritical exponents and the critical value q c .
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.
Critical properties of the qqq-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for qc{q}_{c}qc​, the largest value of qqq for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
Many condensed matter systems, ranging from adsorbed surfaces to bulk magnets, are microscopically modelled by interacting q-state Potts spins, arrayed in d dimensions. A changeover from second-order phase transitions at q≤qc(d) to first-order transitions at q≳qc can be understood as a condensation of effect vacancies, which are patches of local disorder favored by entropy. Accordingly, the renormalization-group treatment of Potts models is within context of Potts-lattice-gas models, where critical and tricritical fixed points occur at low q, but merge and annihilate at qc. This picture has led to exact tricritical exponents in two dimensions. It is also consistent with recent experimental results on intercalated systems in three dimensions. Effective vacancies in pure Potts models have also been studied by Monte Carlo simulation. Their effective chemical potential can be controlled by a four-point interaction, which proved useful in Monte Carlo renormalization-group studies.
The critical behavior of the sss-state Ashkin-Teller-Potts model on a square lattice is studied for s=3,4,and5s=3, 4, \mathrm{and} 5s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the s=3s=3s=3 and s=4s=4s=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=5s=5s=5, however, the transformation fails to detect the first-order transition that is known to occur.