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Physical review. B, Condensed matter

Percolation renormalization-group approach to theq-state Potts model

Chin‐Kun Hu, Chi-Ning Chen

Published Aug 1, 1988• 18 citations
Publisher

Abstract

Based on the connection between the q-state Potts model (QPM) and the q-state bond-correlated percolation model (QBCPM), we have proposed percolation renormalization-group (PRG) methods to calculate the free energy, the critical point, and critical exponents for the QPM. Our methods are free from inconsistency in Larsson's method. We have carried out the ${\ensuremath{\lambda}}_{1}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{1}$ to ${\ensuremath{\lambda}}_{2}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{2}$ renormalization-group (RG) transformations in our methods for several values of original cell sizes ${\ensuremath{\lambda}}_{1}$ and final cell sizes ${\ensuremath{\lambda}}_{2}$ with various boundary conditions. We find that such RG transformations with large ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ and periodic boundary condition usually give accurate physical quantities for the QPM. The advantages and generalization of our approach are also discussed.

DOI
10.1103/physrevb.38.2765

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Journal of Physics A Mathematical and General•Oct 2, 1998•11 citations
Based on the connection between the q-state Potts model (QPM) and the q-state bond-correlated percolation model (QBCPM), we have proposed percolation renormalization-group (PRG) methods to calculate the free energy, the critical point, and critical exponents for the QPM. Our methods are free from inconsistency in Larsson's method. We have carried out the \ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​\ifmmode\times\else\texttimes\fi{}\ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​ to \ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​\ifmmode\times\else\texttimes\fi{}\ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​ renormalization-group (RG) transformations in our methods for several values of original cell sizes \ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​ and final cell sizes \ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​ with various boundary conditions. We find that such RG transformations with large \ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​ and \ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​ and periodic boundary condition usually give accurate physical quantities for the QPM. The advantages and generalization of our approach are also discussed.
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.
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.
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 yt{y}_{t}yt​ for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.
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.
Critical properties of the QQQ-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of QQQ that annihilate as QQQ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For Q>QcQ>{Q}_{c}Q>Qc​, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients aaa and bbb cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, a=6.52a=6.52a=6.52 and b=0.065b=0.065b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, Q=1Q=1Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
The determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.