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International Journal of Modern Physics C

AGING IN A ONE-DIMENSIONAL EDWARDS–ANDERSON SPIN GLASS MODEL WITH LONG-RANGE INTERACTIONS

Marcelo A. Montemurro, Francisco A. Tamarit

Published Mar 1, 2003• 2 citations
Publisher

Abstract

In this work we study, by means of numerical simulations, the out-of-equilibrium dynamics of the one-dimensional Edwards–Anderson model with long-range interactions of the form ± Jr-α. In the limit α → 0 we recover the well known Sherrington–Kirkpatrick mean-field version of the model, which presents a very complex dynamical behavior. At the other extreme, for α → ∞ the model converges to the nearest-neighbor one-dimensional system. We focus our study on the dependence of the dynamics on the history of the sample (aging phenomena) for different values of α. The model is known to have mean-field exponents already for values of α = 2/3. Our results indicate that the crossover to the dynamic mean-field occurs at a value of α < 2/3.

DOI
10.1142/s0129183103004449

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We discuss the dynamics of the infinite-range Sherrington-Kirkpatrick spin-glass model for which relaxation times diverge when NNN, the number of spins, tends to infinity. Calculations on a large but finite system are very difficult, so we mimic a large finite system in equilibrium by working with $N=\ensuremath{\infty}$ and imposing, by hand, a canonical distribution at an initial time. For short times, where no barrier hopping has occurred, we find that the Edwards-Anderson order parameter, qEA{q}_{\mathrm{EA}}qEA​, is identical to that obtained from an analysis of the mean-field equations of Thouless, Anderson, and Palmer and, with further assumptions, gives q(x=1)q(x=1)q(x=1) in Parisi's theory, in agreement with earlier work. For times longer than the longest relaxation time (of the finite system), true equilibrium is reached and our theory agrees with previous statistical-mechanics calculations using the replica trick. There is no violation of the fluctuation-dissipation theorem.
Abstract. Numerical results for the local field distributions of a family of Ising spinglass models are presented. In particular, the Edwards-Anderson model in dimensions two, three, and four is considered, as well as spin glasses with long-range powerlaw-modulated interactions that interpolate between a nearest-neighbour Edwards-Anderson system in one dimension and the infinite-range Sherrington-Kirkpatrick model. Remarkably, the local field distributions only depend weakly on the range of the interactions and the dimensionality, and show strong similarities except for near zero local field. PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+qLocal field distributions in spin glasses 2 1.
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.
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