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arXiv:cond-mat/9909325 · Modern Physics Letters B

THE CRITICAL EXPONENT θ′ IN SPIN GLASSES

H. J. Luo, Lothar Schülke, Bo Zheng

Published Jun 10, 1999• 12 citations
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Abstract

Short-time dynamic scaling behavior of the 3D ± J Ising spin glass is studied by Monte Carlo methods. Starting the replicas with independent initial configurations with a small pseudo-magnetization, the dynamic evolution of the overlap q(t) between two replicas is measured. The initial increase of the overlap q(t) is observed and the corresponding exponent θ′ is obtained. From the scaling relation λ= d/z-θ′, the dynamic exponent z is estimated.

DOI
10.1142/s021798499900052x

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•Jan 1, 2003•1 citation
Short-time dynamic scaling behavior of the 3D ± J Ising spin glass is studied by Monte Carlo methods. Starting the replicas with independent initial configurations with a small pseudo-magnetization, the dynamic evolution of the overlap q(t) between two replicas is measured. The initial increase of the overlap q(t) is observed and the corresponding exponent θ′ is obtained. From the scaling relation λ= d/z-θ′, the dynamic exponent z is estimated.
We use a nonequilibrium Monte Carlo simulation method and dynamical scaling to study the phase transition in three-dimensional Ising spin glasses. The transition point is repeatedly approached at finite velocity v (temperature change versus time) in Monte Carlo simulations starting at a high temperature. This approach has the advantage that the equilibrium limit does not have to be strictly reached for a scaling analysis to yield critical exponents. For the dynamic exponent we obtain z=5.85(9) for bimodal couplings distribution and z=6.00(10) for the Gaussian case. Assuming universal dynamic scaling, we combine the two results and obtain z=5.93±0.07 for generic 3D Ising spin glasses.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.
We have studied numerically the remanent magnetization in the six- and eight-dimensional Ising spin glass and we have compared it with the behaviour observed in the SK model, that we have also computed analytically. We also report the value of the dynamical critical exponent z in six dimensions measured in three different ways: from the behaviour of the energy and the susceptibility as a function of the Monte Carlo time and by studying the overlap - overlap correlation function as a function of space and time. These three results are in very good agreement with the mean field prediction z = 4. Finally we have checked numerically the analytical prediction, obtained by assuming spontaneously broken replica symmetry, for the most singular part of the propagator in the spin-glass phase. This last result supports the existence of spontaneously broken replica symmetry in finite-dimensional spin glasses.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behaviour of 2- and 3-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behaviour of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For 3-dimensional Ising Model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.