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Journal of Physics C Solid State Physics

The field and temperature dependences of the dynamic susceptibility of infinite-range Ising spin glasses

Konrad H. Fischer, Wolfgang Kinzel

Published Sep 10, 1984• 17 citations
Publisher

Abstract

The dynamic susceptibility chi ( omega , T, H) is calculated for an Ising spin glass with infinite-range interactions for small frequencies, omega , and a large range of temperatures and fields. The observed sharp increase of the imaginary part Im chi (T) near the freezing temperature T f for small and vanishing fields is explained by the crossover from Im chi varies as omega at high temperatures to Im chi varies as omega 1/2 at T f . At low temperatures, Im chi increases in proportion to T.

DOI
10.1088/0022-3719/17/25/010

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The dynamic susceptibility chi ( omega , T, H) is calculated for an Ising spin glass with infinite-range interactions for small frequencies, omega , and a large range of temperatures and fields. The observed sharp increase of the imaginary part Im chi (T) near the freezing temperature T f for small and vanishing fields is explained by the crossover from Im chi varies as omega at high temperatures to Im chi varies as omega 1/2 at T f . At low temperatures, Im chi increases in proportion to T.
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By means of two different methods in a Monte Carlo simulation, the authors calculate the AC susceptibilities chi ( omega ) and relaxation time distributions g( tau ) in the +or-J Ising two-dimensional (2D) and three-dimensional (3D) spin glasses. The data obtained by the two methods separately are confirmed to be consistent with each other through the fluctuation dissipation theorem chi ( omega )=(1/T) integral d tau g( tau )/(1-i omega tau ). The imaginary parts Im chi of the AC susceptibilities show a characteristic difference in the 2D and 3D spin glasses; as the frequency of the AC field is decreased, the height of the peak of Im chi becomes higher in the 2D spin glass, and lower in the 3D spin glass. They can also see a difference between the g( tau ) distributions in the 2D and 3D spin glasses, reflected by the difference in the chi ( omega )-values. In the 3D spin glass, Im chi has no frequency dependence at low temperatures, suggesting that the fluctuation spectrum of the magnetization shows 1/f-noise behaviour in the spin-glass phase.
Aging of two-dimensional (2D) Ising spin glasses with short-range interactions Rb2{\mathrm{Rb}}_{2}Rb2​${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$Cox{\mathrm{Co}}_{\mathit{x}}Cox​F4{\mathrm{F}}_{4}F4​ (Tc{\mathit{T}}_{\mathit{c}}Tc​=0) is studied in the critical regime via the low-frequency ac susceptibility and via the dc magnetization for x=0.22 and 0.33. After quenching to low temperatures, slow logarithmic relaxation is observed for both the in-phase and out-of-phase ac susceptibilities. Using the droplet scaling model, the spin-glass correlation length R is concluded to increase after a temperature quench as R\ensuremath{\propto}(lnt${)}^{1/\mathrm{\ensuremath{\psi}}}$, with t in units of some microscopic time. The exponent \ensuremath{\psi}=1.0\ifmmode\pm\else\textpm\fi{}0.1 compares with recent theoretical results and numerical simulations for 2D short-range Ising spin glasses. The waiting-time dependence found for the field-cooled and zero-field-cooled dc magnetizations points to a markedly slower crossover from the quasiequilibrium to the nonequilibrium regime than in 3D. From experiments with different cooling histories, finally, the equilibrium spin-glass state is inferred to be strongly dependent on the temperature.
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A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
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.