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Cluster Percolation and First Order Phase Transitions in the Potts Model

Santo Fortunato, Helmut Satz

Published Aug 6, 2001• 4 citations
OpenAlex

Abstract

The q-state Potts model can be formulated in geometric terms, with Fortuin-Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class. (C) 2002 Published by Elsevier Science B.V.

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The q-state Potts model can be formulated in geometric terms, with Fortuin-Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class. (C) 2002 Published by Elsevier Science B.V.
The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Löwner evolution (SLE κ ) with diffusivity of κ=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross-sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.
The critical behaviour of the Ising model in the absence of an external magnetic field can be specified either through spontaneous symmetry breaking (thermal criticality) or through cluster percolation (geometric criticality). We extend this to finite external fields for the case of the Potts' model, showing that a geometric analysis leads to the same first order/second order structure as found in thermodynamic studies. We calculate the Kertész line, separating percolating and non-percolating regimes, both analytically and numerically for the Potts model in presence of an external magnetic field.
We study the three-point correlation function of the backbone in the two-dimensional Q-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the biconnected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large Q, we employ large-scale Monte Carlo simulations of the O(n) loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with Q=n^{2}. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone (R_{BB}) and FK clusters (R_{FK}). Our computed R_{FK} exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that R_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{FK} holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.
The q-state Potts model on a square lattice and in the presence of an external magnetic field that couples to only one state is studied by using extensive Monte Carlo simulations. A hybrid algorithm that combines the single spin-flip Metropolis method with Wolff cluster updates is employed. Single-histogram reweighting techniques and the field-mixing approach are also used to obtain the corresponding universal probability distributions. By analyzing the finite-size scaling behavior of the order parameters and universal probability distributions, the phase diagram in the temperature-versus-external field plane is obtained in the thermodynamic limit for some values of state q≥5. For positive fields and higher values of q, there is a clear first-order transition line that starts at the zero-field transition point and ends at an isolated critical point, which is in the two-dimensional Ising universality class. As the value of state q decreases, the corresponding critical field also decreases. The present results strongly suggest that the critical field vanishes when q→5^{+}. However, for negative fields, another first-order transition line occurs without any critical or multicritical behavior for q≥6. The special case with q=5 undergoes a weak first-order transition at zero field and a line of second-order transitions for negative fields. This second-order transition line is in the same universality class as the q=4 Potts model.
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.