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arXiv:1411.5512 · Physical Review B

Aging at the spin-glass/ferromagnet transition: Monte Carlo simulations using graphics processing units

Markus Manssen, Alexander K. Hartmann

Published May 26, 2015• 18 citations
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Abstract

We study the nonequilibrium aging behavior of the $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson model in three dimensions for samples of size up to $N={128}^{3}$ and for up to ${10}^{8}$ Monte Carlo sweeps. In particular we are interested in the change of the aging when crossing from the spin-glass phase to the ferromagnetic phase. The necessary long simulation times are reached by employing a CUDA-based GPU implementation, which allows for single-spin flip times as small as 8 ps. We measure typical spin-glass correlation functions in space and time to determine the growing length scale and extract the constituting exponents. We observe a clear signature of the disorder-driven equilibrium transition in the nonequilibrium behavior.

DOI
10.1103/physrevb.91.174433

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We study the nonequilibrium aging behavior of the \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ Edwards-Anderson model in three dimensions for samples of size up to N=1283N={128}^{3}N=1283 and for up to 108{10}^{8}108 Monte Carlo sweeps. In particular we are interested in the change of the aging when crossing from the spin-glass phase to the ferromagnetic phase. The necessary long simulation times are reached by employing a CUDA-based GPU implementation, which allows for single-spin flip times as small as 8 ps. We measure typical spin-glass correlation functions in space and time to determine the growing length scale and extract the constituting exponents. We observe a clear signature of the disorder-driven equilibrium transition in the nonequilibrium behavior.
The crossover between quasiequilibrium and nonequilibrium spin-glass dynamics has been studied in Monte Carlo simulations of two- and three-dimensional short-range Ising spin-glass systems. The spin system was quenched in zero field to a low temperature T. After equilibrating for a time tw{\mathit{t}}_{\mathit{w}}tw​, a weak magnetic field was applied and the time dependence of the magnetization M(t) and the spin autocorrelation function q(t) was studied. Our results show that the relaxation rate S(t)=\ensuremath{\partial}M(t)/\ensuremath{\partial} log10{\mathrm{log}}_{10}log10​(t) exhibits a maximum at t\ensuremath{\approxeq}tw{\mathit{t}}_{\mathit{w}}tw​. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (ta{\mathit{t}}_{\mathit{a}}ta​) of the system, as defined by the maximum in relaxation rate, is shifted to longer or shorter time scales, respectively. No significant differences between the aging behavior in two and three dimensions are found. The relation \ensuremath{\chi}(t)=[1-q(t)]/T holds only for short time scales (t\ensuremath{\ll}tw{\mathit{t}}_{\mathit{w}}tw​), clear deviations are found at longer time scales (t>tw{\mathit{t}}_{\mathit{w}}tw​). Since this relation should according to the fluctuation-dissipation theorem hold for a system in equilibrium, this result is interpreted as an indication of a crossover between equilibrium and nonequilibrium dynamics. Our results are qualitatively in agreement with experiments as well as with theoretical models for the nonequilibrium spin-glass dynamics.
We numerically study aging for the Edwards-Anderson model in three and four dimensions using different temperature-change protocols. In D=3D=3D=3, time scales a thousand times larger than in previous work are reached with the Spin Update Engine (SUE) machine. Deviations from cumulative aging are observed in the nonmonotonic time behavior of the coherence length. Memory and rejuvenation effects are found in a temperature-cycle protocol, revealed by vanishing effective waiting times. Similar effects are reported for the D=3D=3D=3 site-diluted ferromagnetic Ising model (without chaos). However, rejuvenation is reduced if off-equilibrium corrections to the fluctuation-dissipation theorem are considered. Memory and rejuvenation are quantitatively describable in terms of the growth regime of the spin-glass coherence length.
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
Spin-glass (SG) states of the 3-dimensional Ising Edwards-Anderson model under a static magnetic field hhh are examined by means of the standard Monte Carlo simulation on the field-shift aging protocol at temperature TTT. For each process with $(T; \tw, h)$, $\tw$ being the waiting time before the field is switched on, we extract the dynamical crossover time, $\tcr(T; \tw, h)$. We have found a nice scaling relation between the two characteristic length scales which are properly determined from $\tcr$ and $\tw$ and then are normalized by the static field crossover length introduced in the SG droplet theory. This scaling behavior implies the instability of the SG phase in the equilibrium limit even under an infinitesimal hhh. In comparison with this numerical result the field effect on real spin glasses is also discussed.
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.