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Journal of Physics A Mathematical and General

Critical exponents of two-dimensional Potts and bond percolation models

Henk W. J. Blöte, M. P. Nightingale, Bernard Derrida

Published Feb 1, 1981• 85 citations
Publisher

Abstract

Critical exponents of the two-dimensional, q-state Potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q. Results for the temperature exponent agree accurately with the conjecture of den Nijs (1979). The magnetic exponent is found to behave in accordance with the conjecture of Nienhuis et al. (1980).

DOI
10.1088/0305-4470/14/2/005

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Critical exponents of the two-dimensional, q-state Potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q. Results for the temperature exponent agree accurately with the conjecture of den Nijs (1979). The magnetic exponent is found to behave in accordance with the conjecture of Nienhuis et al. (1980).
The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be y T =1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (y T =1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3 / 2 , the difference is nevertheless substantial.
Based on the connection between a q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), we propose that in the calculation of the free energy and a physical quantity, e.g., the magnetic susceptibility, we need only to retain the terms of the most probable cluster-size distribution (MPCSD) defined in the text. For the one-dimensional model, the MPCSD may be calculated exactly. The free energy and the magnetic susceptibility determined by such MPCSD are the same as those calculated exactly by the transfer-matrix method. For the QPM on the lattice with dimensions d\ensuremath{\ge}2, the above assumption about the MPCSD implies that the magnetic susceptibility of the QPM is proportional to the mean cluster sizes of the QBCPM for both T>Tc{T}_{c}Tc​ and TTc{T}_{c}Tc​. The hypothesis on the MPCSD may be extended to other spin models. The implication of this work on the theory of the scaling laws of the critical exponents is also discussed.
We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.
A finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature and critical exponent of the symmetric and the asymmetric two-layer three-state Potts models. For similar intralayer interactions our calculation of the shift exponent $\ensuremath{\varphi}$ confirm some scaling arguments which predict $\ensuremath{\varphi}=\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is the susceptibility exponent. For unequal intralayer interactions we have obtained $\ensuremath{\varphi}=0.5$ which differ from the prediction $\ensuremath{\varphi}=\ensuremath{\gamma}∕2$ of a generalized mean-field theory.
Critical properties of two-dimensional classical models and (1 + 1)-dimensional quantum models are obtained by solving a general dimerized spin chain. The staggered-electric exponent of the FFF model is derived. The central role of umklapp scattering in controlling renormalization and determining Potts-model critical exponents is pointed out. The thermal exponent of the Potts model is evaluated, and the validity of den Nijs conjecture is established. Logarithmic singularities at the multicritical point are obtained.