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Physical review. B, Condensed matter

Critical exponents and marginality of the four-state Potts model: Monte Carlo renormalization group

Robert H. Swendsen, David Andelman, A. Nihat Berker

Published Dec 1, 1981• 30 citations
Publisher

Abstract

The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as ${\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01$, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.

DOI
10.1103/physrevb.24.6732

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The thermal critical exponent of the four-state Potts model in two dimensions is evaluated as \ensuremathν\ensuremath−1=1.49\ifmmode±\else\textpm\fi0.01{\ensuremath{\nu}}^{\ensuremath{-}1}=1.49\ifmmode\pm\else\textpm\fi{}0.01\ensuremathν\ensuremath−1=1.49\ifmmode±\else\textpm\fi0.01, using the Monte Carlo renormalization-group method. The presence of a marginal scaling direction is also indicated. These results confirm previous conjectures, universality, and logarithmic corrections. A purely Potts Hamiltonian is considered. A four-point interaction is used to control the chemical potential of effective vacancies, without the introduction of explicit vacancy states.
The critical behavior of the three-state Potts model in two dimensions is investigated with the use of the Monte Carlo renormalization group (MCRG), and the results are compared with conjectured values as well as with other methods. Recent advances in the MCRG enable greatly improved convergence, as well as greater insight into the structure of the renormalization transformation. The introduction of a four-point interaction is used to control the chemical potential of effective vacancies in a purely Potts Hamiltonian.
A theory of the scaling behavior near the qc=4{q}_{c}=4qc​=4 state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for qqq near qc{q}_{c}qc​. For the pure Potts model with q=qcq={q}_{c}q=qc​ we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At Tc{T}_{c}Tc​ we find ln(r)\mathrm{ln}(r)ln(r) corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with q=qcq={q}_{c}q=qc​ we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as TTT approaches the multicritical point from the first-order side. Results for q>qcq>{q}_{c}q>qc​ and q<qcq<{q}_{c}q<qc​ are also given.
The authors study the critical exponents, eigenoperators and the phase diagram of the N=2 Ashkin-Teller (AT) model in two dimensions using a Monte Carlo renormalisation group. They pay special attention to the marginal operator responsible for the non-universal behaviour. The crossover exponent is measured and used to numerically calculate the thermal exponent of the eight-vertex model and of the AT model near the Potts-four-point, giving better results than a direct determination, since the crossover operator is not affected by the additional AT marginal operators.
The qqq-state Potts models in two dimensions are studied with the Monte Carlo renormalization-group method. The results are in agreement with known properties. Combining our data with the qualitative picture of Nienhuis, Berker, Riedel, and Schick, we obtain a consistent description of the behavior of these models, including nearly singular behavior for q>4q>4q>4 (as proposed by Kim and Joseph) and suggesting logarithmic corrections to scaling for q=4q=4q=4.