arXiv:1105.1902 · Physical Review B

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

Alain H. Billoire, Ian A. Campbell

Published Aug 12, 2011• 15 citations

Abstract

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.

DOI: 10.1103/physrevb.84.054442

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arXiv:1105.1902 · Physical Review B

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

Alain H. Billoire, Ian A. Campbell

Published Aug 12, 2011• 15 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.84.054442

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Numerical study of the dynamics of some long range spin glass models

Alain H. Billoire

We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.

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N D Mackenzie, A. P. Young

The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.

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Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin glass

Cécile Monthus, Thomas Garel

The long-range one-dimensional Ising spin glass with random couplings decaying as $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ presents a spin-glass phase ${T}_{c}(\ensuremath{\sigma})>0$ for $0\ensuremath{\le}\ensuremath{\sigma}<1$ (the limit $\ensuremath{\sigma}=0$ corresponds to the mean-field Sherrington-Kirkpatrick model). We use the eigenvalue method introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. 2009, P12017) to measure the equilibrium time ${t}_{eq}(N)$ at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number $N$ of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.The long-range one-dimensional Ising spin glass with random couplings decaying as $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ presents a spin-glass phase ${T}_{c}(\ensuremath{\sigma})>0$ for $0\ensuremath{\le}\ensuremath{\sigma}<1$ (the limit $\ensuremath{\sigma}=0$ corresponds to the mean-field Sherrington-Kirkpatrick model). We use the eigenvalue method introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. 2009, P12017) to measure the equilibrium time ${t}_{eq}(N)$ at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number $N$ of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.

arXiv:1309.2154Physical Review BJan 9, 201411 citations

A New Method for Studying the Dynamics of Spin Glasses

R. N. Bhatt, A. P. Young

Europhysics Letters (EPL)Sep 1, 199226 citations

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}

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.

N

and temperature

T

. Proper care is taken that the thermal fluctuations on the relaxation time estimates are totally negligible compared to the disorder induced fluctuations. The probability distribution of

\lnτ-\overline{\lnτ}

scales with the scaling variable

N^{1/3} (T_c-T)

strengthening the belief that

\overline{\lnτ}\propto N^{1/3}

in the whole spin glass phase.

{t}_{eq}(N)

at temperature $T={T}_{c}(\ensuremath{\sigma})/2$ as a function of the number

N

of spins. We find the activated scaling $\overline{ln{t}_{eq}(N)}\ensuremath{\sim}{N}^{\ensuremath{\psi}}$ with the same barrier exponent $\ensuremath{\psi}\ensuremath{\simeq}0.33$ in the whole region $0\ensuremath{\le}\ensuremath{\sigma}<1$.