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Journal of Physics C Solid State Physics

Dynamics of q-state Potts models at the percolation threshold

S. Jain, Eduardo Lage, Robin B. Stinchcombe

Published Dec 10, 1986• 7 citations
Publisher

Abstract

The critical dynamics of the q-state Potts model at the percolation threshold is studied using some of the techniques that have been applied recently to diluted Ising systems. Various formulations of single-spin-flip dynamics are considered and shown to lead to a breakdown of conventional dynamic scaling. Furthermore, the predictions are checked by performing Monte Carlo simulations of the two-dimensional three-state Potts model at the bond percolation threshold.

DOI
10.1088/0022-3719/19/34/002

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Journal of Physics A Mathematical and General•Dec 7, 1993•2 citations
The critical dynamics of the q-state Potts model at the percolation threshold is studied using some of the techniques that have been applied recently to diluted Ising systems. Various formulations of single-spin-flip dynamics are considered and shown to lead to a breakdown of conventional dynamic scaling. Furthermore, the predictions are checked by performing Monte Carlo simulations of the two-dimensional three-state Potts model at the bond percolation threshold.
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.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.