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Journal de Physique I

Confirmation of Rieger definition of Ising clusters

Uwe Gropengießer

Published Oct 1, 1994• 1 citation
Publisher

Abstract

A very simple definition of Ising clusters was introduced by Rieger which joins spins with an average individual magnetization opposite to the global magnetization. Involving percolation methods we get very good agreements between the critical temperatures and the percolation thresholds of diluted and undiluted Ising systems in various dimensions.

DOI
10.1051/jp1:1994108

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A very simple definition of Ising clusters was introduced by Rieger which joins spins with an average individual magnetization opposite to the global magnetization. Involving percolation methods we get very good agreements between the critical temperatures and the percolation thresholds of diluted and undiluted Ising systems in various dimensions.
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.
In this paper we consider an approach, which allows researching a processes of order-disorder transition in various systems (with any distribution of the exchange integrals signs) in the frame of Ising model. A new order parameters, which can give a description of a phase transitions, are found. The common definition of these order parameters is the mean value of percolation cluster size. Percolation cluster includes spins with given energy. The transition from absolute disorder to correlated phase could be studied with using of percolation theory methods.
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.
The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including SU(N)\mathrm{SU}(N)SU(N) lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.
We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.