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Physical Review E

Backbone exponents of the two-dimensionalq-state Potts model: A Monte Carlo investigation

Youjin Deng, Henk W. J. Blöte, Bernard Nienhuis

Published Feb 26, 2004• 54 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.69.026114

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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.
We investigate random-cluster representations of the q=1 - and 2-state Potts models in three dimensions, i.e., the bond-percolation and the Ising model, respectively. Using a recently developed sampling technique, we determine the probabilities C1 (r) and C2 (r) that a pair of lattice sites at a distance r are connected by at least one and two mutually independent paths, respectively. The scaling behavior of C1 and C2 at criticality is governed by the magnetic and the backbone scaling dimension X(h) and X(b) , respectively. From a finite-size analysis of the numerical data, we determine X(h) =0.4768 (7) and X(b) =1.125 (3) for the percolation and X(h) =0.5178 (7) and X(b) =0.829 (4) for the Ising model.
The fractal dimensions of the largest clusters of like spins connected by nearest-neighbor bonds in the critical two-dimensional Q-state Potts model have been determined via Monte Carlo simulations to be df{\mathit{d}}_{\mathit{f}}df​=1.946(3), 1.922(5), and 1.909(8) for Q=2, 3, and 4, respectively. These dimensions are close to the magnetic tricritical exponents of the (Q-1)-state dilute Potts model (d-x=187/96=1.947..., 77/40=1.925, and 40/21=1.904... for Q-1=1, 2, and 3, respectively). The critical Q-state model and the tricritical (Q-1)-state model are related to a two-component Potts lattice gas; this model generates percolative susceptibilities of critical Potts clusters, and suggests a renormalization-group argument for df{\mathit{d}}_{\mathit{f}}df​=d-x.
We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.
We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional QQQ-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents dB=1.7320(3){d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)dB​=1.7320(3) and 1.794(2) for Q=2,3Q=2,3Q=2,3, respectively. However, for large QQQ, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n)\mathrm{O}(n)O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n2Q={n}^{2}Q=n2. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops dB=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6){d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)dB​=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+3,4Q=1,2,3,2+\sqrt{3},4Q=1,2,3,2+3​,4, respectively, and dmin=1.0945(2),1.0675(3),1.0475(3),1.0322(4){d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)dmin​=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+3,4Q=2,3,2+\sqrt{3},4Q=2,3,2+3​,4, respectively. Our estimates significantly improve over the existing results for both dB{d}_{B}dB​ and dmin{d}_{\mathrm{min}}dmin​. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of nnn for the leading correction exponent.
The analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field H s besides the bulk one, H B . Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta 2/f delta H s delta H B approximately L -1/8 (1-C'lnL) 15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0).
Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager's critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}=(187/96 and a percolative susceptibility exponent \ensuremath{\gamma}=(91/48, in good agreement with existing numerical estimates. This d\ifmmode\bar\else\textasciimacron\fi{} is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent yJ{\mathrm{y}}_{\mathrm{J}}yJ​=(13/24 controlling the crossover between clusters and droplets.