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Journal of Physics A Mathematical and General

Dynamics of a hierarchical spin glass

Rosane Riera, John Hertz

Published Jun 7, 1991• 2 citations
Publisher

Abstract

The authors study the statics and long-time dynamics of a spin glass on a family of hierarchical lattices. The statics may be solved exactly on these lattices, and they treat the dynamics in a physically motivated low-temperature approximation, retaining only the slowest mode at each renormalization step. The results are generally in good agreement with Monte Carlo simulations in two and three dimensions. In particular, they find dynamical scaling at the spin glass transition in three dimensions, with a dynamical exponent z=6.5. The spin autocorrelation function at T g decays like t -x with x=0.05. A new result for statics which is used in the dynamics calculation is that for the magnetic exponent y ch at the critical point, which is found to be 2.65. Above T g , the autocorrelation function can be fitted in a wide intermediate time range by a Kohlrausch form, but in this model this is only a consequence of a crossover in the renormalization flow and has no fundamental dynamical significance.

DOI
10.1088/0305-4470/24/11/028

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The authors study the statics and long-time dynamics of a spin glass on a family of hierarchical lattices. The statics may be solved exactly on these lattices, and they treat the dynamics in a physically motivated low-temperature approximation, retaining only the slowest mode at each renormalization step. The results are generally in good agreement with Monte Carlo simulations in two and three dimensions. In particular, they find dynamical scaling at the spin glass transition in three dimensions, with a dynamical exponent z=6.5. The spin autocorrelation function at T g decays like t -x with x=0.05. A new result for statics which is used in the dynamics calculation is that for the magnetic exponent y ch at the critical point, which is found to be 2.65. Above T g , the autocorrelation function can be fitted in a wide intermediate time range by a Kohlrausch form, but in this model this is only a consequence of a crossover in the renormalization flow and has no fundamental dynamical significance.
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.
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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.
A spin-glass transition has been investigated for a long time but we have not reached a conclusion yet due to difficulties in the simulation studies. They are slow dynamics, strong finite-size effects, and sample-to-sample dependencies. We found that a size of the spin-glass order reaches a lattice boundary within a very short Monte Carlo step. A competition between the spin-glass order and a boundary condition causes these difficulties. Once the boundary effect was removed, physical quantities exhibited quite normal behaviors. They became self-averaging in a limit of large replica numbers. These findings suggest that the nonequilibrium relaxation method is a good choice for solving the difficulties if a lattice size and a replica number are set sufficiently large. A dynamic scaling analysis on nonequilibrium relaxation functions gave a result that the spin-glass transition and the chiral-glass transition occurs at the same temperature in the Heisenberg model in three dimensions. The estimated critical exponent ν agrees with the experimental result.