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Physical review. B, Condensed matter

Nonlocal percolation in an antiferromagnetic Potts model

Henry Phillip Fried, Michael Schick C

Published Jul 1, 1988• 15 citations
Publisher

Abstract

We devise an algorithm to determine the critical site-occupation probability, ${x}_{c}$, at which antiferromagnetic order in the three-state Potts model first propagates on a triangular lattice. We find ${x}_{c}=0.735$, larger than the value of 0.5 necessary to percolate ferromagnetic order. Even though the means to propagate the order in this problem can be nonlocal as well as local, our results for critical exponents indicate that the percolation transition is in the same universality class as that of ferromagnetic percolation.

DOI
10.1103/physrevb.38.954

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We devise an algorithm to determine the critical site-occupation probability, xc{x}_{c}xc​, at which antiferromagnetic order in the three-state Potts model first propagates on a triangular lattice. We find xc=0.735{x}_{c}=0.735xc​=0.735, larger than the value of 0.5 necessary to percolate ferromagnetic order. Even though the means to propagate the order in this problem can be nonlocal as well as local, our results for critical exponents indicate that the percolation transition is in the same universality class as that of ferromagnetic percolation.
We introduce a new class of site-percolation models and study them by Monte Carlo methods. These percolation models are defined as the zero-temperature limit of a quench-diluted spin model whose symmetries are those of the antiferromagnetic three-state Potts model on a triangular lattice. Many of these percolation models exhibit a sequence of two percolation transitions. A new phase, excluded in the absence of dilution, is found at intermediate spin concentrations. This phase, which we call the chiral phase, exhibits the symmetries of only a subgroup of the full symmetry group of the Potts model. The critical properties of the transitions found in these percolation models have been studied. The results indicate that, for this entire class of percolation models, the observed transitions are all contained in the universality class of the geometric percolation (q=1 Potts) model. From this result we conclude that the breaking of symmetry, which is relevant for thermally driven transitions, is not relevant for percolation transitions.
We use Monte Carlo methods to study the three-state Potts model on a two-dimensional lattice with fivefold coordination. The lattice consists of triangular and hexagonal units and has properties intermediate between the triangular and kagom\'e lattices. The critical behavior of both ferromagnetic and antiferromagnetic cases is considered. As expected, the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. A critical end point is found at the coexistence between the ferromagnetic and antiferromagnetic phases, and the transition to the antiferromagnetic state is always first order.
We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely, K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer-matrix calculations and Monte Carlo simulations. Our investigation covers the whole (K,J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K<0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J→±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.
We investigate the crossover properties of the frustrated percolation model on a two-dimensional square lattice, with asymmetric distribution of ferromagnetic and antiferromagnetic interactions. We determine the critical exponents ν, γ, and β of the percolation transition of the model, for various values of the density of antiferromagnetic interactions π in the range 0≤π≤0.5. Our data is consistent with the existence of a crossover from random percolation behavior for π=0, to frustrated percolation behavior, characterized by the critical exponents of the ferromagnetic 1/2-state Potts model, as soon as π>0.
The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.
We use Monte Carlo method to study three-state Potts model on maple leaf lattice with pure three-site interaction. The critical behavior of both ferromagnetic and antiferromagnetic cases is studied. Our results confirm that the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. For the antiferromagnetic case the transition is of the first order. We have calculated the energy jump and critical temperature in this area. We find there is a tricritical point separating the first order and second order phases for this system.