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Physical review. B, Condensed matter

Bond percolation on a finite lattice: Critical behavior of the (1+ε)-state Potts model

Joseph Rudnick, Jacob Morris, George Gaspari

Published Jan 1, 1994• 1 citation
Publisher

Abstract

The continuous n-state Potts model is reinvestigated. The partition function is cast in a form allowing the limit, n\ensuremath{\rightarrow}1, to be taken prior to perturbation-theoretical expansions. Proceeding in this way, we free the theory of pathologies known to plague other approaches. Several alternatives are presented for a Feynman-diagrammatic expansion of the theory, two of which are identical to those utilized in previous analyses of the mdoel. One of the introduced diagrammatic approaches is used to recover known results of one-loop renormalization calculations. It is also utilized to calculate the partition function of the (1+\ensuremath{\epsilon})-state Potts model, and, hence, the generating function for bond percolation, on a finite lattice. This generating function for cluster statistics exhibits the scaling features proposed by Privman and Fisher for systems of finite size.

DOI
10.1103/physrevb.49.1016

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The continuous n-state Potts model is reinvestigated. The partition function is cast in a form allowing the limit, n\ensuremath{\rightarrow}1, to be taken prior to perturbation-theoretical expansions. Proceeding in this way, we free the theory of pathologies known to plague other approaches. Several alternatives are presented for a Feynman-diagrammatic expansion of the theory, two of which are identical to those utilized in previous analyses of the mdoel. One of the introduced diagrammatic approaches is used to recover known results of one-loop renormalization calculations. It is also utilized to calculate the partition function of the (1+\ensuremath{\epsilon})-state Potts model, and, hence, the generating function for bond percolation, on a finite lattice. This generating function for cluster statistics exhibits the scaling features proposed by Privman and Fisher for systems of finite size.
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.
One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the n+1 n + 1 states Potts model in the limit of n \to 0 n → 0 . In this paper, we show that it is possible to recover the \epsilon ϵ -expansion for critical exponents in finite dimension directly using the M M -layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the M M -layer method to systems where the underlying field theory is unknown or disputed.
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.
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].
We consider the q-state Potts model with correlated random interactions such that their correlations decay at large distances as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$. Applying the renormalization group and a double (\ensuremath{\epsilon},${\ensuremath{\epsilon}}_{a}$) expansion, we derive the scaling form of the equation of state to linear order in \ensuremath{\epsilon}=6-d and ${\ensuremath{\epsilon}}_{a}$=a-2. In the limit q\ensuremath{\rightarrow}1, which describes correlated percolation, the critical amplitudes ${A}^{\ifmmode\pm\else\textpm\fi{}}$ associated with the mean number of clusters F\ensuremath{\sim}${A}^{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\Vert}t${\ensuremath{\Vert}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ are shown to diverge when \ensuremath{\alpha} approaches a negative integer as a function of ${\ensuremath{\epsilon}}_{a}$.
We establish an intriguing connection between geometry and thermodynamics in the critical q-state Potts model on two-dimensional lattices, using the q-state bond-correlated percolation model (QBCPM) representation. We find that the number of clusters <N(c)> of the QBCPM has an energy-like singularity for q not equal to 1, which is reached and supported by exact results, numerical simulation, and scaling arguments. We also establish that the finite-size correction to the number of bonds, <N(b)>, has no constant term and explains the divergence of related quantities as q-->4, the multicritical point. Similar analyses are applicable to a variety of other systems.