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Journal of Physics Condensed Matter

AC susceptibilities and relaxation time distributions for the two-dimensional and three-dimensional +or-J Ising spin glasses

Marina Suzuki, Takayuki Shirakura, Sakari Inawashiro

Published May 6, 1991• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/3/18/008

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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.
We present the results of two sets of numerical simulations of 3-d Ising spin-glasses in the presence of an uniform magnetic field. In the first set, among other quantities, we compute the spin-spin overlap probability distribution P (q ), the link-link overlap probability distribution P e(qe) and the spin-glass susceptibility χ SG for different volumes and temperatures. The results are in good agreement with mean-field behaviour : P(q) and P e(qe) are non trivial and non self-averaging and χSG shows an increase with the linear size L of the system as χSG ∼ Lω. Our estimate is ω = 1,8 ± 0.25 at T = 0.83. In the second set of simulations a small coupling ε is introduced between two copies of the system and the copy overlap Q(ε) is computed as a function of ε. Q(ε) becomes steeper around ε ∼ 0 as L increases, in agreement with the previous set of simulations. Our data seem anyhow incompatible with the alternatives to mean-field theory proposed so far.
Non-exponential slow dynamics in the `Griffiths phase' of the three-dimensional (3D) ± J Ising spin-glass model is studied by means of Monte Carlo simulation. The distribution of relaxation times averaged over samples, \bar P ( x ) with x = lnτ, is calculated from spin auto-correlation functions simulated. It is found that \bar P ( x ) consists of two branches. The cross-over value between the branches, x c , diverges as T → T c , T c being the spin-glass transition temperature. Therefore τ c (= e x c ) is regarded as the critical relaxation time associated with the onset of the spin-glass order at T c , and the branch of \bar P ( x ) at x < x c is interpreted due to the ordinary critical slowing down process ( critical dynamics ). The other branch at x > x c causes the unusual slow dynamics peculiar to spin glasses. Various aspects of \bar P ( x ) of this branch examined are consistent with the following picture; compact regions with less frustrations than the system as a whole are thermally fluctuating with very small frequencies, and independently of each others ( cluster dynamics ).
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.
Spin-glass (SG) and chiral-glass (CG) orderings in three-dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size-scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a lower temperature, which is compatible with zero. Our results of the anisotropic case support the chirality scenario for the phase transitions in the 3D Heisenberg spin glasses.