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Europhysics Letters (EPL)

Exact Relations Between Damage Spreading and Thermodynamical Properties

Antonio Coniglio A, L. de Arcangelis, Hans Jürgen Herrmann, Naeem Jan

Published Feb 15, 1989• 117 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/8/4/003

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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 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.
In this work we extend a recently proposed prescription for studying the dynamical properties of quantum Heisenberg ferromagnets which is based on the spreading of the damage technique. Within the Handscomb Monte Carlo method, we introduce a Hamming distance defined as a difference in the cyclic structure of Mayers diagrams in such a way that the quantum nature of the Hamiltonian is accounted for. We show that the cyclic Hamming distance is closely related to the static spin susceptibility and illustrate our results by comparing them with previous two-loop renormalization group calculations of the susceptibility low-temperature behavior in two-dimensional quantum Heisenberg models.
We apply the spreading of the damage technique to a S=1/2 Ising ferromagnetic thin film. We use Metropolis dynamics to calculate the temperature dependence of the Hamming distance as well as the Hamming distance distributions along the Markov chain. We suggest a modification of the algorithm for calculating the Hamming distance in the case of Metropolis dynamics and we discuss the results for the two-dimensional model obtained by the modified approach. For a film with three layers we present the thermal dependences of the above quantities. We also suggest a new way to introduce damage to the system as the difference in the boundary conditions. We discuss the results of this new approach for the two-dimensional model.
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.
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.