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Journal of Physics A Mathematical and General

Damage spreading in the Potts and Ashkin-Teller models: exact results

Ananias M. Mariz

Published Mar 21, 1990• 30 citations
Publisher

Abstract

Following the approach introduced by Coniglio et al. (1989) for the Ising model, the author studies the relation between damage spreading and thermal properties in the q-state Potts model as well as in the Ashkin-Teller model. The author shows that by selecting appropriate combinations of different types of damage (q(q-1) in the Potts model and 12 in the Ashkin-Teller model), it is possible to find exact relations between damage and thermodynamic functions, like the magnetisation and the pair correlation function.

DOI
10.1088/0305-4470/23/6/019

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“Damage spreading” is a useful tool for determining equilibrium thermal properties from Monte Carlo simulations of Ising models. Formal exact relations relate static equilibrium properties, e.g. the correlation function, to the equilibrium damage. Similar exact relations also relate the time-dependent correlation function to the time-dependent damage. However, some results such as the direct determination of the characteristic time, τ, from damage spreading appear to be at odds with that reported in the literature by more traditional models. We show that some of these discrepancies, but not all, may be resolved by taking the appropriate scaling function into account.

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Following the approach introduced by Coniglio et al. (1989) for the Ising model, the author studies the relation between damage spreading and thermal properties in the q-state Potts model as well as in the Ashkin-Teller model. The author shows that by selecting appropriate combinations of different types of damage (q(q-1) in the Potts model and 12 in the Ashkin-Teller model), it is possible to find exact relations between damage and thermodynamic functions, like the magnetisation and the pair correlation function.
Some exact results are derived relating quantities computable by the so-called damage spreading method and thermodynamic functions for the N -color Ashkin–Teller model. The results are valid for any ergodic dynamics. Since we restrict our analysis to the ferromagnetic case, the results are also valid for any translationally invariant lattice. The relations derived can be used in order to determine the N -color Ashkin–Teller critical exponents numerically, providing better accuracy and requiring less computational effort than standard Monte Carlo simulations.
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.
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.
“Damage spreading” is a useful tool for determining equilibrium thermal properties from Monte Carlo simulations of Ising models. Formal exact relations relate static equilibrium properties, e.g. the correlation function, to the equilibrium damage. Similar exact relations also relate the time-dependent correlation function to the time-dependent damage. However, some results such as the direct determination of the characteristic time, τ, from damage spreading appear to be at odds with that reported in the literature by more traditional models. We show that some of these discrepancies, but not all, may be resolved by taking the appropriate scaling function into account.