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Physical Review B

Percolation in one of q colors near criticality

Xiaofeng Qian, Youjin Deng, Henk W. J. Blöte

Published Apr 29, 2005• 13 citations
Publisher

Abstract

We study bond percolation in two dimensions between random site variables having one out of $q$ colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature $T$ in the disordered Potts range ${T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}$ for several $q$-state Potts Hamiltonians. For high $T$, these transitions fit, irrespective of $q$, in the universality class of the ordinary percolation transitions. However, for $T\ensuremath{\downarrow}{T}_{c}$, $q$-dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at $q=2$. For $q<2$ the Potts critical state appears to enhance percolation, for $q>2$ it appears to suppress it. Remarkably, for $q=2$ the percolation line coincides with the only flow line extending to $T>{T}_{c}$ from the critical fixed point associated with Potts clusters.

DOI
10.1103/physrevb.71.144303

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We study bond percolation in two dimensions between random site variables having one out of qqq colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature TTT in the disordered Potts range Tc\ensuremath⩽T<\ensuremath∞{T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}Tc​\ensuremath⩽T<\ensuremath∞ for several qqq-state Potts Hamiltonians. For high TTT, these transitions fit, irrespective of qqq, in the universality class of the ordinary percolation transitions. However, for T\ensuremath↓TcT\ensuremath{\downarrow}{T}_{c}T\ensuremath↓Tc​, qqq-dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at q=2q=2q=2. For q<2q<2q<2 the Potts critical state appears to enhance percolation, for q>2q>2q>2 it appears to suppress it. Remarkably, for q=2q=2q=2 the percolation line coincides with the only flow line extending to T>TcT>{T}_{c}T>Tc​ from the critical fixed point associated with Potts clusters.
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.
Simulations of an Ising qqq-state Potts model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for q=2,q=2,q=2, there are two critical Ising lines meeting at T=0T=0T=0 at the four-state Potts critical point; (ii) for 2<q< 4,2<q<~4,2<q< 4, the Ising critical line meets the qqq-state critical line and a line of first-order transitions at a bicritical point; (iii) for q>4,q>4,q>4, the Ising critical line intersects a line of first-order transitions at a critical end point.
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point Tc{T}_{c}Tc​ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability pB<1,{p}_{B}<1,pB​<1, the corresponding site-bond clusters keep on percolating at Tc{T}_{c}Tc​ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability pc{p}_{c}pc​ such that, for any pB> pc,{p}_{B}>~{p}_{c},pB​> pc​, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for pc<pB< 1{p}_{c}<{p}_{B}<~1pc​<pB​< 1 but, for pB=pc,{p}_{B}{=p}_{c},pB​=pc​, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.
Universality is a fundamental concept in modern physics. For the q-state Potts model, the critical exponents are merely determined by the order-parameter symmetry S_{q}, spatial dimensionality and interaction range, independent of microscopic details. In a simplest and mean-field treatment, i.e., the Potts model on complete graph (CG), the phase transition is further established to be of percolation universality for the range of 0<q<2. By simulating the CG Potts model in the random-cluster representation, we numerically demonstrate such a hyperuniversality that the critical exponents are the same for 0<q<2 and, moreover, the Ising system (q=2) exhibits a variety of critical geometric properties in percolation universality. On the other hand, many other universal properties in the finite-size scaling (FSS) theory, including Binder-like ratios and distribution function of the order parameter, are observed to be q dependent. Meanwhile, we have made improvements to the Monte Carlo algorithms for efficiently simulating the CG Potts model. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.
We study the effect of varying strength $\ensuremath{\delta}$ of bond randomness on the phase transition of the three-dimensional Potts model for large qqq. The cooperative behavior of the system is determined by large correlated domains in which the spins point in the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder $\ensuremath{\delta}>{\ensuremath{\delta}}_{t}$ this percolating cluster coexists with a percolating cluster of noncorrelated spins. Such a coexistence is only possible in more than two dimensions. We argue and check numerically that ${\ensuremath{\delta}}_{t}$ is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.10(2)$ and ${\ensuremath{\nu}}_{t}=0.67(4)$. We claim these exponents are qqq independent for sufficiently large qqq. In the second-order transition regime the critical exponents ${\ensuremath{\beta}}_{t}∕{\ensuremath{\nu}}_{t}=0.60(2)$ and ${\ensuremath{\nu}}_{t}=0.73(1)$ are independent of the strength of disorder.