Physical Review Letters

Statics and Dynamics of a Two-Dimensional Ising Spin-Glass Model

A. P. Young

Published Mar 21, 1983• 111 citations

Abstract

The temperature and field dependence of spatial correlations and relaxation times are investigated in detail by Monte Carlo simulations for a two-dimensional Ising spin-glass model. There is no transition, but, in zero field, barrier heights and correlation range increase smoothly at low temperatures. This increase does not seem to be fast enough to explain experiments. In a field, barrier heights and the correlation length tend to a finite limit as $T\ensuremath{\rightarrow}0$. Points in the $h\ensuremath{-}T$ plane with constant relaxation time satisfy $T(h)\ensuremath{-}T(0)\ensuremath{\propto}{h}^{\frac{2}{3}}$ at moderately low temperatures.

DOI: 10.1103/physrevlett.50.917

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Physical Review Letters

Statics and Dynamics of a Two-Dimensional Ising Spin-Glass Model

A. P. Young

Published Mar 21, 1983• 111 citations

Publisher

Abstract

T\ensuremath{\rightarrow}0

. Points in the

h\ensuremath{-}T

plane with constant relaxation time satisfy

T(h)\ensuremath{-}T(0)\ensuremath{\propto}{h}^{\frac{2}{3}}

at moderately low temperatures.

DOI: 10.1103/physrevlett.50.917

Related papers

Critical behavior of three-dimensional Ising model of spin glass (invited)

Andrew T. Ogielski, I. Morgenstern

First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.First results of massive Monte Carlo simulations of the d=3 Ising spin glass with ± J bond distribution, performed on a fast special purpose computer, are discussed. Equilibrium static and dynamic correlation functions have been recorded for temperatures T/J=1.10 to 2.00. A qualitative change in the behavior of the system, and the best fits for the spin glass correlation length and relaxation time favor a continuous phase transition at Tc/J≊1.20±0.05. We find no evidence for the Tc=0 freezing picture.

Journal of Applied PhysicsApr 15, 198521 citations

Aging in two-dimensional Ising spin glasses

A. G. Schins, E. M. Dons, A. F. M. Arts +4 more

Aging of two-dimensional (2D) Ising spin glasses with short-range interactions ${\mathrm{Rb}}_{2}$ ${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ ${\mathrm{Co}}_{\mathit{x}}$ ${\mathrm{F}}_{4}$ ( ${\mathit{T}}_{\mathit{c}}$ =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.Aging of two-dimensional (2D) Ising spin glasses with short-range interactions ${\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Co}}_{\mathit{x}}$${\mathrm{F}}_{4}$ (${\mathit{T}}_{\mathit{c}}$=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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J.-O. Andersson, Johan Mattsson, Peter Svedlindh

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 ${\mathit{t}}_{\mathit{w}}$ , 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} ${\mathrm{log}}_{10}$ (t) exhibits a maximum at t\ensuremath{\approxeq} ${\mathit{t}}_{\mathit{w}}$ . If the temperature is lowered or raised, immediately prior to the field application, the apparent age ( ${\mathit{t}}_{\mathit{a}}$ ) 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} ${\mathit{t}}_{\mathit{w}}$ ), clear deviations are found at longer time scales (t> ${\mathit{t}}_{\mathit{w}}$ ). 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.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 ${\mathit{t}}_{\mathit{w}}$, 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} ${\mathrm{log}}_{10}$(t) exhibits a maximum at t\ensuremath{\approxeq}${\mathit{t}}_{\mathit{w}}$. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (${\mathit{t}}_{\mathit{a}}$) 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}${\mathit{t}}_{\mathit{w}}$), clear deviations are found at longer time scales (t>${\mathit{t}}_{\mathit{w}}$). 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.

Physical review. B, Condensed matterOct 1, 199265 citations

Monte Carlo Study on the 2D ±JIsing Spin Glass –Dynamical Behavior of Individual Spins and Spin Clusters–

Koji Nemoto, Hiroshi Matsukawa, Hajime Takayama

We study the Monte Carlo simulation on the two-dimensional ± J Ising spin glass aiming to clarify the relation between behavior of individual spins and spin clusters in each sample and common spin glass properties of the system. Particularly relaxation times of individual spins are evaluated and their temperature dependences are examined. Making use of data obtained, we discuss in detail spin-freezing process in this model for spin glass. One of our most interesting findings is that spin relaxation processes at temperatures $T{\gtrsim}0.8J$ obey Fulcher's law with T 0 ≃0.5 J , implying that the characteristic relaxation time of the system may diverge at T 0 .We study the Monte Carlo simulation on the two-dimensional ± J Ising spin glass aiming to clarify the relation between behavior of individual spins and spin clusters in each sample and common spin glass properties of the system. Particularly relaxation times of individual spins are evaluated and their temperature dependences are examined. Making use of data obtained, we discuss in detail spin-freezing process in this model for spin glass. One of our most interesting findings is that spin relaxation processes at temperatures $T{\gtrsim}0.8J$ obey Fulcher's law with T 0 ≃0.5 J , implying that the characteristic relaxation time of the system may diverge at T 0 .

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Dynamics of three-dimensional Ising spin glasses in thermal equilibrium

Andrew T. Ogielski

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)=〈 ${S}_{x}$ (0) ${S}_{x}$ (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 ( ${8}^{3}$ , ${16}^{3}$ , ${32}^{3}$ , and ${64}^{3}$ ). 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.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)=〈${S}_{x}$(0)${S}_{x}$(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 (${8}^{3}$, ${16}^{3}$, ${32}^{3}$, and ${64}^{3}$). 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.

Physical review. B, Condensed matterDec 1, 1985773 citations

T\ensuremath{\rightarrow}0

. Points in the

h\ensuremath{-}T

plane with constant relaxation time satisfy

T(h)\ensuremath{-}T(0)\ensuremath{\propto}{h}^{\frac{2}{3}}

at moderately low temperatures.

Aging of two-dimensional (2D) Ising spin glasses with short-range interactions

{\mathrm{Rb}}_{2}

${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$

{\mathrm{Co}}_{\mathit{x}}

{\mathrm{F}}_{4}

(

{\mathit{T}}_{\mathit{c}}

=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.

{\mathit{t}}_{\mathit{w}}

, 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}

{\mathrm{log}}_{10}

(t) exhibits a maximum at t\ensuremath{\approxeq}

{\mathit{t}}_{\mathit{w}}

. If the temperature is lowered or raised, immediately prior to the field application, the apparent age (

{\mathit{t}}_{\mathit{a}}

) 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}

{\mathit{t}}_{\mathit{w}}

), clear deviations are found at longer time scales (t>

{\mathit{t}}_{\mathit{w}}

). 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.

T{\gtrsim}0.8J

obey Fulcher's law with T 0 ≃0.5 J , implying that the characteristic relaxation time of the system may diverge at T 0 .

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)=〈

{S}_{x}

(0)

{S}_{x}

(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 (

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

). 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.