Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations. Spin glasses (SG) exhibit characteristic slow dynamics below the SG transition temperature. 1)-3) Recently such slow dynamics, in particular aging phenomena, has been much attractive both in experimental and theoretical studies. Several attempts have been made so far to explain the slow dynamics. There are mainly two distinctAging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations. Spin glasses (SG) exhibit characteristic slow dynamics below the SG transition temperature. 1)-3) Recently such slow dynamics, in particular aging phenomena, has been much attractive both in experimental and theoretical studies. Several attempts have been made so far to explain the slow dynamics. There are mainly two distinct

Jan 1, 19998 citations

Numerical Study on Aging Dynamics in the 3D Ising Spin-Glass Model. II. Quasi-Equilibrium Regime of Spin Auto-Correlation Function

達雄小森, Hajime Yoshino, Hajime Takayama

Using Monte Carlo simulations, we have studied isothermal aging of three-dimensional Ising spin-glass model focusing on quasi-equilibrium behavior of the spin auto-correlation function. Weak violation of the time translational invariance in the quasi-equilibrium regime is analyzed in terms of {\it effective stiffness} for droplet excitations in the presence of domain walls. Within the range of computational time window, we have confirmed that the effective stiffness follows the expected scaling behavior with respect to the characteristic length scales associated with droplet excitations and domain walls, whose growth law has been extracted from our simulated data. Implication of the results are discussed in relation to experimental works on ac susceptibilities.Using Monte Carlo simulations, we have studied isothermal aging of three-dimensional Ising spin-glass model focusing on quasi-equilibrium behavior of the spin auto-correlation function. Weak violation of the time translational invariance in the quasi-equilibrium regime is analyzed in terms of {\it effective stiffness} for droplet excitations in the presence of domain walls. Within the range of computational time window, we have confirmed that the effective stiffness follows the expected scaling behavior with respect to the characteristic length scales associated with droplet excitations and domain walls, whose growth law has been extracted from our simulated data. Implication of the results are discussed in relation to experimental works on ac susceptibilities.

arXiv:cond-mat/9908078Journal of the Physical Society of JapanApr 15, 200058 citations

NUMERICAL STUDY ON AGING DYNAMICS IN THE 3D ISING SPIN-GLASS MODEL — QUASI-EQUILIBRIUM BEHAVIOR OF SPIN AUTO-CORRELATION FUNCTIONS

H. TAKAYAMA, Hajime Yoshino, T. KOMORI

Using Monte Carlo simulations, we have studied aging phenomena in three-dimensional Gaussian Ising spin-glass model focusing on quasi-equilibrium behavior of the spin auto-correlation functions. Weak violation of the time translational invariance in the quasi-equilibrium regime is analyzed in terms of effective stiffness for droplet excitations in the presence of domain walls. The simulation results not only in isothermal aging but also in T-shift aging process. T-shift aging processes exhibit the expected scaling behavior with respect to the characteristic length scales associated with droplet excitations and domain walls in spite of the fact that the growth law for these length scales still shows a pre-asymptotic behavior compared with the asymptotic form proposed by the droplet theory. Implications of our simulational results are also discussed in relation to experimental observations.Using Monte Carlo simulations, we have studied aging phenomena in three-dimensional Gaussian Ising spin-glass model focusing on quasi-equilibrium behavior of the spin auto-correlation functions. Weak violation of the time translational invariance in the quasi-equilibrium regime is analyzed in terms of effective stiffness for droplet excitations in the presence of domain walls. The simulation results not only in isothermal aging but also in T-shift aging process. T-shift aging processes exhibit the expected scaling behavior with respect to the characteristic length scales associated with droplet excitations and domain walls in spite of the fact that the growth law for these length scales still shows a pre-asymptotic behavior compared with the asymptotic form proposed by the droplet theory. Implications of our simulational results are also discussed in relation to experimental observations.

arXiv:cond-mat/9909228International Journal of Modern Physics CDec 1, 19993 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.

Physical review. B, Condensed matterDec 1, 199318 citations

Monte Carlo studies of Ising spin-glass systems: Aging behavior and crossover between equilibrium and nonequilibrium dynamics

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

Equilibrium behavior of the spin-glass ordered phase

Daniel S. Fisher, David A. Huse

A phenomenological theory of the ordered phase of short-range Ising spin glasses is developed in terms of droplet excitations and presented in detail. These excitations have free energies with a broad distribution whose characteristic magnitude grows with length scale L as ${L}^{\ensuremath{\theta}}$. A small fraction of droplets of all scales are thermally active; these dominate much of the physics. The mean-square correlation functions are found to decay with distance as 1/${r}^{\ensuremath{\theta}}$ for all T ${T}_{c}$ and the autocorrelations decay logarithmically with time because of large activation barriers for creation and annihilation of the droplet excitations. A renormalization procedure is sketched in order to define excitations at positive temperature. It is found that the long-distance equilibrium correlation functions are extremely sensitive to small temperature changes, yielding breakdown of certain relations between fluctuations and thermodynamic derivatives. The behavior near to the critical temperature is discussed and some of the ideas are extended to systems with power-law interactions and to spin glasses with X-Y or Heisenberg symmetry. The inequality \ensuremath{\theta}\ensuremath{\le}(d-1)/2 is also derived.A phenomenological theory of the ordered phase of short-range Ising spin glasses is developed in terms of droplet excitations and presented in detail. These excitations have free energies with a broad distribution whose characteristic magnitude grows with length scale L as ${L}^{\ensuremath{\theta}}$. A small fraction of droplets of all scales are thermally active; these dominate much of the physics. The mean-square correlation functions are found to decay with distance as 1/${r}^{\ensuremath{\theta}}$ for all T${T}_{c}$ and the autocorrelations decay logarithmically with time because of large activation barriers for creation and annihilation of the droplet excitations. A renormalization procedure is sketched in order to define excitations at positive temperature. It is found that the long-distance equilibrium correlation functions are extremely sensitive to small temperature changes, yielding breakdown of certain relations between fluctuations and thermodynamic derivatives. The behavior near to the critical temperature is discussed and some of the ideas are extended to systems with power-law interactions and to spin glasses with X-Y or Heisenberg symmetry. The inequality \ensuremath{\theta}\ensuremath{\le}(d-1)/2 is also derived.

Physical review. B, Condensed matterJul 1, 1988925 citations

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}_{c}

and the autocorrelations decay logarithmically with time because of large activation barriers for creation and annihilation of the droplet excitations. A renormalization procedure is sketched in order to define excitations at positive temperature. It is found that the long-distance equilibrium correlation functions are extremely sensitive to small temperature changes, yielding breakdown of certain relations between fluctuations and thermodynamic derivatives. The behavior near to the critical temperature is discussed and some of the ideas are extended to systems with power-law interactions and to spin glasses with X-Y or Heisenberg symmetry. The inequality \ensuremath{\theta}\ensuremath{\le}(d-1)/2 is also derived.