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Analytic and computational studies of spin glasses

Shreyans Jain

Published Jan 1, 1985• 0 citations

Abstract

The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.

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Role of initial conditions in the mean-field theory of spin-glass dynamics

A. Houghton, S. Jain, A. P. Young

We discuss the dynamics of the infinite-range Sherrington-Kirkpatrick spin-glass model for which relaxation times diverge when $N$, the number of spins, tends to infinity. Calculations on a large but finite system are very difficult, so we mimic a large finite system in equilibrium by working with $N=\ensuremath{\infty}$ and imposing, by hand, a canonical distribution at an initial time. For short times, where no barrier hopping has occurred, we find that the Edwards-Anderson order parameter, ${q}_{\mathrm{EA}}$, is identical to that obtained from an analysis of the mean-field equations of Thouless, Anderson, and Palmer and, with further assumptions, gives $q(x=1)$ in Parisi's theory, in agreement with earlier work. For times longer than the longest relaxation time (of the finite system), true equilibrium is reached and our theory agrees with previous statistical-mechanics calculations using the replica trick. There is no violation of the fluctuation-dissipation theorem.

Physical review. B, Condensed matter•Sep 1, 1983•56 citations

Statics and dynamics of the infinite-range Ising spin glass model

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.

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We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.

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Jairo Rolim Lopes de Almeida

This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.

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Lack of self-averaging in spin glasses

A. P. Young, Alan J. Bray, M A Moore

The authors show that there are significant sample variations for the probability distribution of the overlap between phases in the Sherrington-Kirkpatrick spin glass model even in the thermodynamic limit. However experimentally observable quantities, such as the magnetisation, energy and Edwards-Anderson (finite time) order parameter take the same value in all samples. The fluctuation-dissipation theorem holds when a sample average is performed but not for each sample separately.

Journal of Physics C Solid State Physics•Feb 20, 1984•43 citations

Role of initial conditions in spin glass dynamics and significance of Parisi's q(x)

A. Houghton, Shreyans Jain, A. P. Young

The authors argue that standard treatments of dynamics are incorrect for non-ergodic systems such as the Sherrington-Kirkpatrick model of a spin glass. Since the system never loses the memory of its initial state it is necessary to average over initial states with a Boltzmann weight to obtain time-dependent correlation functions as conventionally defined. It is now necessary to introduce replicas. The 'dynamical' order parameter, q EA , is equal to q(x=1) in Parisi's scheme whereas the statistical mechanics order parameter, q, is given by integral 0 I q(x) dx. They also propose an interpretation of the function q(x).

Journal of Physics C Solid State Physics•Apr 30, 1983•46 citations

The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.

We discuss the dynamics of the infinite-range Sherrington-Kirkpatrick spin-glass model for which relaxation times diverge when

N

, the number of spins, tends to infinity. Calculations on a large but finite system are very difficult, so we mimic a large finite system in equilibrium by working with $N=\ensuremath{\infty}$ and imposing, by hand, a canonical distribution at an initial time. For short times, where no barrier hopping has occurred, we find that the Edwards-Anderson order parameter,

{q}_{\mathrm{EA}}

, is identical to that obtained from an analysis of the mean-field equations of Thouless, Anderson, and Palmer and, with further assumptions, gives

{q}_{\mathrm{EA}}

in Parisi's theory, in agreement with earlier work. For times longer than the longest relaxation time (of the finite system), true equilibrium is reached and our theory agrees with previous statistical-mechanics calculations using the replica trick. There is no violation of the fluctuation-dissipation theorem.

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.

We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.

This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.

The authors argue that standard treatments of dynamics are incorrect for non-ergodic systems such as the Sherrington-Kirkpatrick model of a spin glass. Since the system never loses the memory of its initial state it is necessary to average over initial states with a Boltzmann weight to obtain time-dependent correlation functions as conventionally defined. It is now necessary to introduce replicas. The 'dynamical' order parameter, q EA , is equal to q(x=1) in Parisi's scheme whereas the statistical mechanics order parameter, q, is given by integral 0 I q(x) dx. They also propose an interpretation of the function q(x).