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arXiv:cond-mat/0205211 · Physical review. B, Condensed matter

Critical droplets and phase transitions in two dimensions

Santo Fortunato

Published Jan 15, 2003• 16 citations
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Abstract

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point ${T}_{c}$ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability ${p}_{B}<1,$ the corresponding site-bond clusters keep on percolating at ${T}_{c}$ 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 ${p}_{c}$ such that, for any ${p}_{B}>~{p}_{c},$ 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 ${p}_{c}<{p}_{B}<~1$ but, for ${p}_{B}{=p}_{c},$ 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.

DOI
10.1103/physrevb.67.014102

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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 pB=pCK=1\ensuremath−exp(\ensuremath−2J/kT):{p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):pB​=pCK​=1\ensuremath−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.
The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including SU(N)\mathrm{SU}(N)SU(N) lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.
The critical behaviour of several spin models can be simply described as percolation of some suitably defined clusters or droplets: the onset of the geometrical transition coincides with the critical point, and the percolation exponents are equal to the thermal exponents. It is still unknown whether, given a model, one can define at all the droplets. In the cases where this is possible, the droplet definition depends in general on the specific model at study and can be quite involved. We propose here a simple general definition for the droplets: they are clusters obtained by joining nearest-neighbour spins of the same sign with some bond probability p CK , which is the minimal probability that still allows the existence of a percolating cluster at the critical temperature T c . This definition has been recently introduced and indeed satisfies the conditions required for the droplets, for many classical spin models, discrete and continuous, in two dimensions. By means of lattice Monte Carlo simulations we show here that the definition works as well in three dimensions. In particular, our prescription allows us to describe exactly the confinement–deconfinement transition of SU ( N ) gauge theories as Polyakov loop percolation.
Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability ppp between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent yp=0.426(6)y_p = 0.426(6)yp​=0.426(6) and the fractal dimensions of the largest cluster, hull, and shortest path as df=1.8926(20)d_f = 1.8926(20)df​=1.8926(20), dhull=1.663(4)d_{\rm hull} = 1.663(4)dhull​=1.663(4), and dmin=1.080(10)d_{\rm min} = 1.080(10)dmin​=1.080(10), respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.
The fractal dimensions and the percolation exponents of geometrical spin clusters of like sign at criticality are obtained numerically for an Ising model with temperature dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D c , and their interfaces D I , are in perfect agreement with those reported for the standard two-dimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles results in a diffusivity κ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.
We propose a site random-cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation. To simulate the model on a square lattice, we combine the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon. To verify whether or not it is consistent with the bond random-cluster model, we measure several quantities, such as the wrapping probability Re, the percolating cluster density P∞, and the magnetic susceptibility per site χp, as well as two exponents, such as the thermal exponent yt and the fractal dimension yh of the percolating cluster. We find that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5, and 4, the numerical estimation of the exponents yt and yh are consistent with the theoretical values. The universalities of the site random-cluster model and the bond random-cluster model are completely identical. For larger values of q, we find obvious signatures of the first-order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site. Our results are helpful for the understanding of the percolation of traditional statistical models.
We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.