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arXiv:0711.1148 · Journal of Physics A Mathematical and Theoretical

On the Kertész line: thermodynamic versus geometric criticality

Philippe Blanchard, D Gandolfo, Lahoussine Laanait, Jean Ruiz, Helmut Satz

Published Feb 12, 2008• 11 citations
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Abstract

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.

DOI
10.1088/1751-8113/41/8/085001

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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.
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 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.
The Q-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general special choices of these magnetic fields produce effective models with smaller symmetry . The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behaviour, for finite values of the symmetry-breaking fields, in the cases where Q > 4 is also analysed. Our results indicate that for effective models with symmetry the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.
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.
Exact analytic results are presented for the critical properties of the dilute Ising model on a Bethe lattice, where a fraction p of the sites are occupied at random by spins. It is found that the critical behaviour is the same as that of the pure system (p=1) except for transitions which occur by varying p at zero temperature, where the critical behaviour is different and corresponds to the percolation problem. For T to 0 and p to p c , the percolation concentration, the zero-field magnetization shows crossover behaviour of the form expected from scaling theories. Outside the critical region, a rapidly convergent numerical method is developed and used to calculate the spontaneous magnetization.
Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), the author defines a non-percolating geometrical factor g f and percolating geometrical factor g p which depend only on q and geometrical properties of the system. The thermal properties of the QPM could be derived simply from g f and g p . The idea is confirmed by an exact calculation for one-dimensional QPM and may be extended to other Ising-like spin models. The formulation also provides a geometrical meaning for the finite-size scaling and broadening at first-order phase transitions of the spin model.