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arXiv:cond-mat/9904303 · Journal of Physics A Mathematical and General

Numerical simulations of the four-dimensional Edwards-Anderson spin glass with binary couplings

Enzo Marinari, Francesco Zuliani

Published Oct 13, 1999• 36 citations
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Abstract

We present numerical results that allow a precise determination of the transition point and of the critical exponents of the four-dimensional Edwards-Anderson spin glass with binary quenched random couplings. We show that the low-temperature phase undergoes replica symmetry breaking. We obtain results on large lattices, up to a volume V = 10 4 : we use finite-size scaling to show the relevance of our results in the infinite volume limit.

DOI
10.1088/0305-4470/32/43/301

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We present numerical results that allow a precise determination of the transition point and of the critical exponents of the four-dimensional Edwards-Anderson spin glass with binary quenched random couplings. We show that the low-temperature phase undergoes replica symmetry breaking. We obtain results on large lattices, up to a volume V = 10 4 : we use finite-size scaling to show the relevance of our results in the infinite volume limit.
We present numerical simulations of the four-dimensional Edwards Anderson Ising spin glass with binary couplings. Our results, in the midst of strong finite-size effects, suggest the existence of a spin-glass phase transition. We present a preliminar determination of critical exponents. We discuss spin-glass susceptibilities, cumulants of the overlap and energy overlap probability distributions, finite-size effects, and the behaviour of the disorder-dependent and integrated probability distributions.
We characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.
By using numerical simulations we show that the four-dimensional Edwards Anderson spin glass in magnetic field undergoes a mean-field-like phase transition. We use a dynamical approach: we simulate large lattices (of volume V ) and work out the behaviour of the system in a limit where both t and V go to infinity, but where the limit is taken first. By showing that the dynamic overlap q converges to a value smaller than the static one we exhibit replica symmetry breaking. The critical exponents are compatible with those obtained by mean-field computations.
The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite-size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the streaming single instruction, multiple data extensions), we have been able to simulate the four-dimensional Edwards-Anderson model with Gaussian couplings up to sizes L=70 and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite-size effects are negligible: ξ(t)≲L/7. Our estimates for the dynamical exponent (z≃1/T) and for the replicon exponent (α≃1.0 and T independent), that controls the decay of the spatial correlation in the zero overlap sector, are consistent with the replica symmetry breaking theory, but the latter differs from the theoretically conjectured value.
We perform numerical simulations, including parallel tempering, a four-state Potts glass model with binary random quenched couplings using the JANUS application-oriented computer. We find and characterize a glassy transition, estimating the critical temperature and the value of the critical exponents. Nevertheless, the extrapolation to infinite volume is hampered by strong scaling corrections. We show that there is no ferromagnetic transition in a large temperature range around the glassy critical temperature. We also compare our results with those obtained recently on the ``random permutation'' Potts glass.