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Physical Review E

Damage-spreading simulations through exact relations for the two-dimensional Potts ferromagnet

Aécia S. D. dos Anjos, D. A. Moreira, Ananias M. Mariz, Fernando Dantas Nobre

Published Jul 20, 2006• 5 citations
Publisher

Abstract

A powerful computational method for dealing with correlation functions in magnetic systems, based on damage-spreading simulations, is reviewed and tested, by investigating the q-state Potts ferromagnet, on a square lattice, at criticality. Exact relations involving special kinds of damage and the spin-spin correlation function, as well as the magnetization, are used. The efficiency of the method arises with a significant reduction of the finite-size effects, with respect to conventional Monte Carlo simulations. Correlation functions, which represent usually a hard task within this latter procedure, appear to be much more easily estimated through the present damage-spreading simulations. The effectiveness of the technique is illustrated by an accurate estimate of the exponent eta, of the spin-spin correlation function, for q=2, 3, and 4, with rather small lattice sizes. In the cases q > or = 5, an analysis of the magnetization is consistent with the well-known first-order phase transition.

DOI
10.1103/physreve.74.016703

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A powerful computational method for dealing with correlation functions in magnetic systems, based on damage-spreading simulations, is reviewed and tested, by investigating the q-state Potts ferromagnet, on a square lattice, at criticality. Exact relations involving special kinds of damage and the spin-spin correlation function, as well as the magnetization, are used. The efficiency of the method arises with a significant reduction of the finite-size effects, with respect to conventional Monte Carlo simulations. Correlation functions, which represent usually a hard task within this latter procedure, appear to be much more easily estimated through the present damage-spreading simulations. The effectiveness of the technique is illustrated by an accurate estimate of the exponent eta, of the spin-spin correlation function, for q=2, 3, and 4, with rather small lattice sizes. In the cases q > or = 5, an analysis of the magnetization is consistent with the well-known first-order phase transition.
The nearest-neighbor-interaction ferromagnetic Ashkin-Teller model is investigated on a square lattice through a powerful computational method for dealing with correlation functions in magnetic systems. This technique, which is based on damage-spreading numerical simulations, makes use of exact relations involving special kinds of damage and correlation functions, as well as the corresponding order parameters of the model. The computation of correlation functions, which represents usually a hard task in standard Monte Carlo simulations, due to large fluctuations, turns out to be much simpler within the present approach. We concentrate our analysis along the Baxter line, well known for its continuously varying critical exponents; seven different points along this line are investigated. The critical exponents associated with correlation functions along the Baxter line are successfully evaluated, by means of numerical methods, within damage-spreading simulations. The efficiency of this method is confirmed through precise estimates of the critical exponents associated with the order parameters (magnetization and polarization), as well as with their corresponding correlation functions, in spite of the small lattice sizes considered.
The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.
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 present a numerical study of two-dimensional random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc=4, which discriminates between second- and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.
The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.
We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2 d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d −β/ν at T c and whose size distribution is also related to static exponents.