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Journal of Physics A Mathematical and General

The effect of dynamics on damage spreading in the two-dimensional classical XY model

J Chiu, Stephen Teitel

Published Sep 7, 1990• 12 citations
Publisher

Abstract

The authors study damage spreading in the classical two-dimensional XY model, using a dynamics and distance measure which preserve the rotational invariance of the Hamiltonian. They find only a high temperature random phase and a low temperature ordered phase, consistent with equilibrium results. Their results contrast to previous results of Golinelli and Derrida (1988).

DOI
10.1088/0305-4470/23/17/008

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Journal of Physics A Mathematical and General•Oct 7, 1989•29 citations
The authors study damage spreading in the classical two-dimensional XY model, using a dynamics and distance measure which preserve the rotational invariance of the Hamiltonian. They find only a high temperature random phase and a low temperature ordered phase, consistent with equilibrium results. Their results contrast to previous results of Golinelli and Derrida (1988).
The time evolution of the Hamming distance (damage spreading) for the and Ising models on the square lattice is performed with a special metropolis dynamics algorithm. Two distinct regimes are observed according to the temperature range for both models: a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size; a high-temperature one where the distance vanishes in the long-time limit. Using the finite size scaling method, the dynamical phase transition (damage spreading transition) temperature is obtained as for the Ising model.
We present results showing the correlation between the out-of-equilibrium dynamics and the equilibrium damage-spreading process in the two-dimensional ± J Edwards–Anderson model at low temperatures. A key ingredient in our analysis is the projection of finite temperature spin configurations onto the ground state topology of the system. In particular, through numerical simulations we correlate ground state information with the out-of-equilibrium dynamics. We also analyse how the propagation of a small perturbation in equilibrated systems is related to the ground state topology. This damage-spreading study unveils the presence of rigid clusters of spins. We claim that these clusters give rise to the slow out-of-equilibrium dynamics observed in the temperature range between the glass temperature T g = 0 of the two-dimensional ± J Edwards–Anderson model and the critical temperature T c of the pure ferromagnetic Ising model.
We investigate the spreading of damage in the three-dimensional Ising model by means of large-scale Monte Carlo simulations. Within the Glauber dynamics we use different rules for the order in which the sites are updated. We find that the stationary damage values and the spreading temperature are different for different update order. In particular, random update order leads to larger damage and a lower spreading temperature than regular order. Consequently, damage spreading in the Ising model is nonuniversal not only with respect to different update algorithms (e.g., Glauber vs heat-bath dynamics) as already known, but even with respect to the order of sites.
The time evolution of the Hamming distance (damage spreading)for the S=1/2 and S=1 Ising models on the square lattice is performed with a special metropolis dynamics algorithm.Two distinct regimes are observed according to the temperature range for both models:a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size;a high- temperature one where the distance vanishes in the long-time limit.Using the finite size scaling method,the dynamical phase transition(damage spreading transition) temperature is obtained as Tc ≈1.675±0.025 for the S=1 lsing model.
The 2D XY model is studied by Monte Carlo simulations where two configurations subjected to the same thermal noise are compared. The authors observe three temperature regimes: a high-temperature phase where the two configurations meet, an intermediate phase where they do not meet but their distance becomes independent of the initial distance and a low-temperature phase where the distance remembers for very long the initial distance. The two transition temperatures T 1 =1.61+or-0.01 and T 2 =0.95+or-0.05 are estimated by means of finite-size scaling.