Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal de Physique I

Self-similar phase space in spin glasses

Serge Galam, A. Mauger

Published Mar 1, 1994• 1 citation
Publisher

Abstract

Postulating homogeneity for the relaxation function, we show that the standard dynamic scaling hypothesis is recovered and that effective relaxation times must decreme upon increasing Kadanoff's blocks sizes. In the case of spin glasses our results are naturally obtained within a diffusion process in the phase space among eigenstates available to the spin system. The model suggests that the critical slowing-down in spin glasses does not result from hierarchically constrained dynamics.

DOI
10.1051/jp1:1994111

Related papers

Dynamic scaling and non exponential relaxations in the presence of disorder. Application to spin glasses

B. Castaing, Jean Souletie

Assuming that hierarchical procedure governs the evolution from one to the next step of Kadanoff's renormalization in the presence of some disorder one finds that the dynamic response is described by a log-normal distribution of the relaxation times. There are two natural limits to the model which restitute either the dynamic scaling or the activated dynamic hypotheses. This provides the basis for a simple justification to the success of the usual fits by a stretched exponential or by a Cole-Cole law which are commonly made to the non-exponential responses observed in the time or in the frequency domains in spin glasses. We provide expressions for the temperature dependences of the parameters which control these laws. We also propose a model for freezing which is based on the idea that $\partial$ ln $\tau/\partial(1/T)$ should be continuous at the freezing point. We discuss available data in spin glasses but our conclusions should be very general.

Journal de Physique I•Mar 1, 1991•30 citations

Non-exponential relaxation near a transition in the presence of disorder: The case of spin glasses

Jean Souletie

Within the framework of a model which associates Kadanoff's renormalization with a hierarchical condition, we justify dynamic scaling (τ/τ ≈ (ξ/ξ0)z) and non-exponential relaxations (m(t) = t −1/zv at T = T C). In the presence of some disorder the same model predicts β → α transition from a short-time regime where we observe activated T In t scaling for short-range effects, to a regime where we have stretched exponential behaviour and dynamic scaling (τ ≈ τ0(1 – T C/T)−zv) for long-range effects.

Philosophical Magazine B•Jun 1, 1992•3 citations

Hierarchical scaling: An analytical approach to slow relaxations in spin glasses, glasses, and other correlated systems (invited)

Jean Souletie

A model has been developed that associates a hierarchical criterion to the usual renormalization of the distances near a phase transition. At times t smaller than τα∝(1−Tc/T)−zν, which diverges at Tc, it is found that the relaxation develops at a scale s(t), which is a power law of the time elapsed: s ∝ tT/zTc. If one sits in the ergodic regime T≳Tc, the system abruptly reaches an equilibrium at s(τα)=ξ(T)=τα1/z. If T<Tc, by contrast, it deviates to logarithmic behavior. The α relaxation is well approximated by a stretched exponential exp(−t/τα)β(T), which traduces the effect of underlying lognormal statistics. The exponent β(T)∝[1−b ln(1−Tc/T)]−1/2 would cancel at Tc if it was possible to wait for the sample to equilibrate at this temperature. The model has the necessary ingredients to describe effects that have been reported in both the ‘‘activated’’ nonergodic regime and in the critical regime of spin glasses, and of other correlated systems like orientational glasses, glass-forming liquids, and vortex glasses. A possible extension of the model is discussed to describe the ‘‘macroscopic quantum effects.’’

Journal of Applied Physics•May 15, 1994•11 citations

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.

Physical review. B, Condensed matter•Dec 1, 1985•773 citations

Evidence for Universal Scaling in the Spin-Glass Phase

Thomas Jörg, Helmut G. Katzgraber

We perform Monte Carlo simulations of Ising spin-glass models in three and four dimensions, as well as of Migdal-Kadanoff spin glasses on a hierarchical lattice. Our results show strong evidence for universal scaling in the spin-glass phase in all three models. Not only does this allow for a clean way to compare results obtained from different coupling distributions, it also suggests that a so far elusive renormalization group approach within the spin-glass phase may actually be feasible.

arXiv:0803.3792•Physical Review Letters•Nov 5, 2008•19 citations

Dynamic Theory of Spin Glasses

有博 富樫, Masuo Suzuki

Here is presented a dynamic theory of spin glasses based on the extended mean-field theory of Thouless, Anderson, and Palmer. The kinetic equations for the magnetization and the spin correlation function are derived, from which dynamical linear and nonlinear susceptibilities are calculated and their low-frequency behavior is mainly investigated. The spin self-correlation functions have different relaxation times near above and below T c –the former is proportional to | T - T c | -2 , while the latter to | T - T c | -4 –which shows the asymmetric critical slowing down. A qualitative comparison of the present results with experimental data is presented.

Journal of the Physical Society of Japan•Sep 1, 1983•2 citations
Postulating homogeneity for the relaxation function, we show that the standard dynamic scaling hypothesis is recovered and that effective relaxation times must decreme upon increasing Kadanoff's blocks sizes. In the case of spin glasses our results are naturally obtained within a diffusion process in the phase space among eigenstates available to the spin system. The model suggests that the critical slowing-down in spin glasses does not result from hierarchically constrained dynamics.
Assuming that hierarchical procedure governs the evolution from one to the next step of Kadanoff's renormalization in the presence of some disorder one finds that the dynamic response is described by a log-normal distribution of the relaxation times. There are two natural limits to the model which restitute either the dynamic scaling or the activated dynamic hypotheses. This provides the basis for a simple justification to the success of the usual fits by a stretched exponential or by a Cole-Cole law which are commonly made to the non-exponential responses observed in the time or in the frequency domains in spin glasses. We provide expressions for the temperature dependences of the parameters which control these laws. We also propose a model for freezing which is based on the idea that ∂\partial∂ ln τ/∂(1/T)\tau/\partial(1/T)τ/∂(1/T) should be continuous at the freezing point. We discuss available data in spin glasses but our conclusions should be very general.
Within the framework of a model which associates Kadanoff's renormalization with a hierarchical condition, we justify dynamic scaling (τ/τ ≈ (ξ/ξ0)z) and non-exponential relaxations (m(t) = t −1/zv at T = T C). In the presence of some disorder the same model predicts β → α transition from a short-time regime where we observe activated T In t scaling for short-range effects, to a regime where we have stretched exponential behaviour and dynamic scaling (τ ≈ τ0(1 – T C/T)−zv) for long-range effects.
A model has been developed that associates a hierarchical criterion to the usual renormalization of the distances near a phase transition. At times t smaller than τα∝(1−Tc/T)−zν, which diverges at Tc, it is found that the relaxation develops at a scale s(t), which is a power law of the time elapsed: s ∝ tT/zTc. If one sits in the ergodic regime T≳Tc, the system abruptly reaches an equilibrium at s(τα)=ξ(T)=τα1/z. If T<Tc, by contrast, it deviates to logarithmic behavior. The α relaxation is well approximated by a stretched exponential exp(−t/τα)β(T), which traduces the effect of underlying lognormal statistics. The exponent β(T)∝[1−b ln(1−Tc/T)]−1/2 would cancel at Tc if it was possible to wait for the sample to equilibrate at this temperature. The model has the necessary ingredients to describe effects that have been reported in both the ‘‘activated’’ nonergodic regime and in the critical regime of spin glasses, and of other correlated systems like orientational glasses, glass-forming liquids, and vortex glasses. A possible extension of the model is discussed to describe the ‘‘macroscopic quantum effects.’’
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.
Here is presented a dynamic theory of spin glasses based on the extended mean-field theory of Thouless, Anderson, and Palmer. The kinetic equations for the magnetization and the spin correlation function are derived, from which dynamical linear and nonlinear susceptibilities are calculated and their low-frequency behavior is mainly investigated. The spin self-correlation functions have different relaxation times near above and below T c –the former is proportional to | T - T c | -2 , while the latter to | T - T c | -4 –which shows the asymmetric critical slowing down. A qualitative comparison of the present results with experimental data is presented.