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Physical review. B, Condensed matter

Percolation, finite-size scaling, and the thermal scaling power for the Potts model

Chin‐Kun Hu, Chi-Ning Chen

Published Jul 1, 1989• 1 citation
Publisher

Abstract

Based on the connection between the q-state Potts model (QPM) and q-state bond-correlated percolation model, a simple geometrical method to calculate the thermal scaling power ${y}_{t}$ for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.

DOI
10.1103/physrevb.40.854

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Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model, Hu has defined the percolating geometrical factor Np{\mathit{N}}_{\mathit{p}}Np​ and the nonpercolating geometrical factors Nf{\mathit{N}}_{\mathit{f}}Nf​. From Np{\mathit{N}}_{\mathit{p}}Np​ nd Nf{\mathit{N}}_{\mathit{f}}Nf​, we may obtain the thermal properties of the QPM and construct a percolation renormalization-group method to study the QPM. In this paper we present a fast algorithm to calculate exact Np{\mathit{N}}_{\mathit{p}}Np​ and Nf{\mathit{N}}_{\mathit{f}}Nf​ for the QPM on finite cells.
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It has been shown that the partition function of the q-state Potts model (QPM) is the generating function of a q-state bond-correlated percolation model (QBCPM) such that the spontaneous magnetization M, the magnetic susceptibility \ensuremath{\chi}, the internal energy U, and the specific heat Ch{C}_{h}Ch​ for the QPM are related to the percolation probability P, the mean cluster size S, the average number of occupied bond p\ifmmode\bar\else\textasciimacron\fi{}, and the fluctuations of occupied bonds F for the QBCPM. We use the Monte Carlo simulation method of Swendsen and Wang and the multiple labeling technique of Hoshen and Kopelman to calculate P, S, p\ifmmode\bar\else\textasciimacron\fi{}, and F for the QBCPM on the square and the simple cubic lattice. Our calculations give accurate critical points for the QPM.
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