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Physical Review Letters

New definition and simulation of critical clusters of Ising spins

Z. Alexandrowicz

Published Feb 22, 1988• 22 citations
Publisher

Abstract

Critical correlation in thermal systems, Ising spins for example, is approximately described by clusters. A cluster is defined as a randomly picked, connected region of excess up (or down) spins. Its simulation utilizes a branching-out stochastic chain which allots spins continuously, either to the cluster or to a perimeter of average composition. Results in dimensionality from 2 to 10 show the cluster geometry to be very different than in (athermal) percolation.

DOI
10.1103/physrevlett.60.669

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Critical correlation in thermal systems, Ising spins for example, is approximately described by clusters. A cluster is defined as a randomly picked, connected region of excess up (or down) spins. Its simulation utilizes a branching-out stochastic chain which allots spins continuously, either to the cluster or to a perimeter of average composition. Results in dimensionality from 2 to 10 show the cluster geometry to be very different than in (athermal) percolation.
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.
We consider geometrical or Ising clusters (i.e., domains of parallel spins) in the square lattice random-field Ising model by varying the strength of the Gaussian random field $\ensuremath{\Delta}$. In agreement with the conclusion of a previous investigation [Phys. Rev. E 63, 066109 (2001)], the geometrical correlation length, i.e., the average size of the clusters $\ensuremath{\xi}$ is finite for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}\ensuremath{\approx}1.65$ and divergent for $\ensuremath{\Delta}\ensuremath{\leqslant}{\ensuremath{\Delta}}_{c}$. The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand, the divergence of the correlation length, $\ensuremath{\xi}(\ensuremath{\Delta})\ensuremath{\sim}{(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c})}^{\ensuremath{-}\ensuremath{\nu}}$, with $\ensuremath{\nu}\ensuremath{\approx}2$, is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.
A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of sss spins is shown to scale as lns\mathrm{ln}slns. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.
We study the percolation properties of geometrical clusters defined in the overlap space of two statistically independent replicas of a square-lattice Ising model that are simulated at the same temperature. In particular, we consider two distinct types of clusters in the overlap, which we dub soft- and hard-constraint clusters, and which are subsets of the regions of constant spin overlap. By means of Monte Carlo simulations and a finite-size scaling analysis we estimate the transition temperature as well as the set of critical exponents characterizing the percolation transitions undergone by these two cluster types. The results suggest that both soft- and hard-constraint clusters percolate at the critical temperature of the Ising model and their critical behavior is governed by the correlation-length exponent ν=1 found by Onsager. At the same time, they exhibit nonstandard and distinct sets of exponents for the average cluster size and percolation strength.
The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized" percolation rules correlated to entire blocks of spins. For example these rules allow for percolation on "virtual" bonds between spins not present in the underlying Hamiltonian. As a consequence we can define new random cluster theories each with its own Monte Carlo cluster dynamics that exactly reproduce the Ising model. By tuning parameters on the critical percolation surface, it is demonstrated numerically that cluster algorithms can be formulated for the 2-d and 3-d Ising model that have smaller autocorrelations than the original Swendsen-Wang algorithm.