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Europhysics Letters (EPL)

Damage Spreading in Spin Glasses

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

Published Aug 15, 1989• 48 citations
Publisher

Abstract

By studying the time evolution of two configurations submitted to the same thermal noise, we investigate phase space properties of the three-dimensional ± J Ising spin glass. Our results for the distance between states are consistent with the picture of phase space having a multivalley structure below T c . By fixing the damage at one site, we also study the Hamming distance between the two configurations at its critical point, where the distribution of probability of getting damaged n times is found to be multifractal.

DOI
10.1209/0295-5075/9/8/002

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Exact Relations Between Damage Spreading and Thermodynamical Properties

Antonio Coniglio A, L. de Arcangelis, Hans Jürgen Herrmann +1 more

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.

Europhysics Letters (EPL)•Feb 15, 1989•117 citations
By studying the time evolution of two configurations submitted to the same thermal noise, we investigate phase space properties of the three-dimensional ± J Ising spin glass. Our results for the distance between states are consistent with the picture of phase space having a multivalley structure below T c . By fixing the damage at one site, we also study the Hamming distance between the two configurations at its critical point, where the distribution of probability of getting damaged n times is found to be multifractal.
We study the time evolution of the distance between two configurations submitted to the same thermal noise for the 3 d ± J Ising spin glass. We observe three temperature regimes: a high-temperature regime where the distances vanishes in the long-time limit. An intermediate-temperature regime where the distance has a nonzero limit independent of the initial distance. A low-temperature regime where the distance in the long time limit seems to depend upon the initial distance. For the sake of comparison, we have repeated our simulations for the ferromagnetic case.
Using information of the ground-state topology we show that the damage spreading technique unveils ferromagnetic order in the three-dimensional ± J Edwards-Anderson spin glass model. With spin-flipping dynamics damage spreads for temperatures larger than T g , the glass transition temperature. With spin-orienting dynamics and for temperatures in T g < T < T d , damage spreads over a finite region of the system, composed of finite clusters of ferromagnetic character. T d is the spin-orienting damage critical temperature, which is of the same order as the critical temperature of the ferromagnetic Ising model, T c . The present results allow for an interpretation within a single framework of known —and sometimes puzzling— results, giving an intuitive picture for growing order in spin glasses.
The dynamics of a mixed spin Ising model with a crystal field on the square lattice are investigated by using the damage spreading technique and Metropolis sampling method.The simulation results reveal that the damage(Hamming distance)shows a very similar temperature dependence to that of the magnetization.The Hamming distance decreases with temperature and there is a critical temperature,beyond which the phase space trajectories of two systems with different initial configurations consist with each other.The critical temperature increases when the strength of the crystal field decreases,and the mixed spin Ising model will degenerate to a standard Ising model under a considerable weak crystal field.The critical temperature decreases to zero continuously as the strength of the crystal filed increases.There is not any first order phase transition observed in the above process and thus no tricritical point exists in this system.
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.