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arXiv:cond-mat/0501198 · Europhysics Letters (EPL)

The mean-field infinite range p = 3 spin glass: Equilibrium landscape and correlation time scales

Alain H. Billoire, L Giomi, Enzo Marinari

Published Jul 28, 2005• 19 citations
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Abstract

We investigate numerically the dynamical behavior of the mean field 3-spin spin glass model: we study equilibrium dynamics, and compute equilibrium time scales as a function of the system size V . We find that for increasing volumes the time scale τ increases like ln τ ∝ V . We also present an accurate study of the equilibrium static properties of the system.

DOI
10.1209/epl/i2005-10149-4

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We investigate numerically the dynamical behavior of the mean field 3-spin spin glass model: we study equilibrium dynamics, and compute equilibrium time scales as a function of the system size V . We find that for increasing volumes the time scale τ increases like ln τ ∝ V . We also present an accurate study of the equilibrium static properties of the system.
I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈Sx{S}_{x}Sx​(0)Sx{S}_{x}Sx​(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.
The statics-dynamics correspondence in spin glasses relate non-equilibrium results on large samples (the experimental realm) with equilibrium quantities computed on small systems (the typical arena for theoretical computations). Here we employ statics-dynamics equivalence to study the Ising spin-glass critical behavior in three dimensions. By means of Monte Carlo simulation, we follow the growth of the coherence length (the size of the glassy domains), on lattices too large to be thermalized. Thanks to the large coherence lengths we reach, we are able to obtain accurate results in excellent agreement with the best available equilibrium computations. To do so, we need to clarify the several physical meanings of the dynamic exponent close to the critical temperature.
For pt.I see ibid., vol.16, no.7, p.1219 (1983). The response of a simple infinite-range model of a spin glass to the imposition of a small external field is studied. The field is turned on at t=0, and the evolution of the spin correlations is followed as processes involving flips of successively larger and larger clusters of spins occur, until equilibrium is achieved at t= infinity . The evolution from one state to the other proceeds through a quasi-continuum of intermediate metastable states labelled by the fraction y of spins that flip simultaneously on that time scale. The time scale for such processes grows very long as the size N of the system increases (possibly like exp((Ny)12/Tg/T)), so in the infinite system, only an infinitesimal fraction of this evolution can be observed. The generalised equation of state describing the evolution of correlations has the form derived by Sompolinsky, but the physical interpretation is different. In particular, the author does not postulate any violation of the fluctuation-dissipation theorem in equilibrium; the terms in the equations which correspond to those which violate it in his theory have the form they do because a non-equilibrium situation is being described. The physical interpretation of the variable y also allows an additional relation between initial- and final-state correlation functions which leads to the Parisi equation for his order parameter q(x), with x=1-y.
Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale tau (T) controls the dynamics. tau (T) diverges upon approaching the T = 0 critical point. The divergence of tau (T -> 0) is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses.
The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.
We investigate by means of Monte Carlo simulations the fully connected p -state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behaviour of the system in the thermodynamic limit. Using p = 10 we are able to study the equilibrium dynamics for system sizes as large as N = 2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P ( q ) show very strong finite-size effects. From P ( q ) we calculate the fourth-order cumulant g 4 ( N , T ) and the Guerra parameter G ( N , T ) and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated. Also the spin-autocorrelation function C ( t ) shows strong finite-size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature T D . We show that the dependence on N and T of the α-relaxation time can be understood by means of a dynamical finite-size scaling ansatz. C ( t ) does not obey the time-temperature superposition principle for temperatures around T D , but does so for significantly lower T . Finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin, i.e. that the system is dynamically very heterogeneous, which explains the non-exponentiality of C ( t ).