Physical review. B, Condensed matter

Ordering and relaxation in spin glasses

Ian A. Campbell

Published Mar 1, 1986• 48 citations

Abstract

Ordering and relaxation in spin glasses are discussed in terms of the diffusion of the point representing the system in eigenstate space. On this approach, the spin-glass transition ${T}_{g}$ appears as a percolation threshold; stretched exponential relaxation behavior extends from ${T}_{g}$ to an upper characteristic temperature ${T}_{c}$. Predictions are consistent with three-dimensional Ising simulations by Ogielski. Experimental data suggest that for real metallic spin glasses ${T}_{c}$\ensuremath{\simeq}1.5${T}_{g}$. This approach may be relevant for other broad-spectrum relaxation phenomena.

DOI: 10.1103/physrevb.33.3587

Related papers

Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above Tg

Alain H. Billoire, Ian A. Campbell

arXiv:1105.1902Physical Review BAug 12, 201115 citations

Nonequilibrium relaxation study of Ising spin glass models

Yukiyasu Ozeki, Nobuyasu Ito

Physical review. B, Condensed matterJun 20, 200136 citations

The experimental status on the question of a phase transition in spin glasses

Gwyn Williams

Canadian Journal of PhysicsOct 1, 19874 citations

Dynamical critical phenomena in three-dimensional Heisenberg spin glasses

Mitsuhiro Matsumoto, Koji Hukushima, Hajime Takayama

arXiv:cond-mat/0204225Physical review. B, Condensed matterSep 4, 200222 citations

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

Andrew T. Ogielski, I. Morgenstern

Journal of Applied PhysicsApr 15, 198521 citations

Low-temperature phase boundary of dilute-lattice spin glasses

Stefan Boettcher, Emiliano Marchetti

Physical Review BMar 27, 20083 citations

Physical review. B, Condensed matter

Ordering and relaxation in spin glasses

Ian A. Campbell

Published Mar 1, 1986• 48 citations

Publisher

Abstract

Ordering and relaxation in spin glasses are discussed in terms of the diffusion of the point representing the system in eigenstate space. On this approach, the spin-glass transition

{T}_{g}

appears as a percolation threshold; stretched exponential relaxation behavior extends from

{T}_{g}

to an upper characteristic temperature

{T}_{c}

. Predictions are consistent with three-dimensional Ising simulations by Ogielski. Experimental data suggest that for real metallic spin glasses

{T}_{c}

\ensuremath{\simeq}1.5

{T}_{g}

. This approach may be relevant for other broad-spectrum relaxation phenomena.

DOI: 10.1103/physrevb.33.3587

Related papers

Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above Tg

Alain H. Billoire, Ian A. Campbell

A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature ${T}_{g}$ . The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at ${T}_{g}$ . The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature ${T}_{g}$. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at ${T}_{g}$. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.

arXiv:1105.1902Physical Review BAug 12, 201115 citations

Nonequilibrium relaxation study of Ising spin glass models

Yukiyasu Ozeki, Nobuyasu Ito

As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The $\ifmmode\pm\else\textpm\fi{}J$ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of $L=127$ and for four-dimensional (hypercubic) lattice up to $L=41.$ These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time ${t}_{\mathrm{w}}$ from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature ${T}_{\mathrm{g}}.$ It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of ${T}_{\mathrm{g}}.$ Together with these two bounds, we propose a new dynamical way for the estimation of ${T}_{\mathrm{g}}$ from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The $\ifmmode\pm\else\textpm\fi{}J$ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of $L=127$ and for four-dimensional (hypercubic) lattice up to $L=41.$ These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time ${t}_{\mathrm{w}}$ from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature ${T}_{\mathrm{g}}.$ It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of ${T}_{\mathrm{g}}.$ Together with these two bounds, we propose a new dynamical way for the estimation of ${T}_{\mathrm{g}}$ from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.

Physical review. B, Condensed matterJun 20, 200136 citations

The experimental status on the question of a phase transition in spin glasses

Gwyn Williams

Following a brief introduction to spin-glass systems, the current status of experimental evidence on the occurrence of a phase transition is reviewed. Recent analyses of dynamical properties of spin glasses are discussed in terms of several model predictions, including the conclusion of a divergent relaxation time τ at a finite temperature, although with large, non-mean-field indices z and v. The role of regular terms (leading to corrections to scaling) in the nonlinear response are reviewed, stressing their influence on the evaluation of the susceptibility and order-parameter exponents γ′ and β′.Following a brief introduction to spin-glass systems, the current status of experimental evidence on the occurrence of a phase transition is reviewed. Recent analyses of dynamical properties of spin glasses are discussed in terms of several model predictions, including the conclusion of a divergent relaxation time τ at a finite temperature, although with large, non-mean-field indices z and v. The role of regular terms (leading to corrections to scaling) in the nonlinear response are reviewed, stressing their influence on the evaluation of the susceptibility and order-parameter exponents γ′ and β′.

Canadian Journal of PhysicsOct 1, 19874 citations

Dynamical critical phenomena in three-dimensional Heisenberg spin glasses

Mitsuhiro Matsumoto, Koji Hukushima, Hajime Takayama

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.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.

arXiv:cond-mat/0204225Physical review. B, Condensed matterSep 4, 200222 citations

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

Low-temperature phase boundary of dilute-lattice spin glasses

Stefan Boettcher, Emiliano Marchetti

The thermal-to-percolative crossover exponent $\ensuremath{\phi}$, well known for ferromagnetic systems, is studied extensively for Edwards--Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold ${p}_{c}$ using an improved reduction algorithm. Simulations extend to system sizes above $N={10}^{8}$ in dimensions $d=2,\dots{},7$ . The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for ${T}_{g}$ provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.The thermal-to-percolative crossover exponent $\ensuremath{\phi}$, well known for ferromagnetic systems, is studied extensively for Edwards--Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold ${p}_{c}$ using an improved reduction algorithm. Simulations extend to system sizes above $N={10}^{8}$ in dimensions $d=2,\dots{},7$. The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for ${T}_{g}$ provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.

Physical Review BMar 27, 20083 citations

Ordering and relaxation in spin glasses are discussed in terms of the diffusion of the point representing the system in eigenstate space. On this approach, the spin-glass transition

{T}_{g}

appears as a percolation threshold; stretched exponential relaxation behavior extends from

{T}_{g}

to an upper characteristic temperature

{T}_{c}

. Predictions are consistent with three-dimensional Ising simulations by Ogielski. Experimental data suggest that for real metallic spin glasses

{T}_{c}

\ensuremath{\simeq}1.5

{T}_{g}

. This approach may be relevant for other broad-spectrum relaxation phenomena.

A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature

{T}_{g}

. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at

{T}_{g}

. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.

L=127

and for four-dimensional (hypercubic) lattice up to

L=41.

These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time

{t}_{\mathrm{w}}

from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature

{T}_{\mathrm{g}}.

It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of

{T}_{\mathrm{g}}.

Together with these two bounds, we propose a new dynamical way for the estimation of

{T}_{\mathrm{g}}

from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.

{p}_{c}

using an improved reduction algorithm. Simulations extend to system sizes above

N={10}^{8}

in dimensions

d=2,\dots{},7

. The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for

{T}_{g}

provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.