spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as

{e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}

for

T

greater than

{T}_{c}

, the spin-glass transition temperature. At and below

{T}_{c}

, spin-spin correlations are observed to decay to their long-time limit as

{t}^{\ensuremath{-}\frac{1}{2}}

DOI: 10.1103/physrevb.17.4384

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Infinite-ranged models of spin-glasses

Scott Kirkpatrick, David Sherrington

A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and $\mathrm{XY}$ spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for $T$ greater than ${T}_{c}$ , the spin-glass transition temperature. At and below ${T}_{c}$ , spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and $\mathrm{XY}$ spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for $T$ greater than ${T}_{c}$, the spin-glass transition temperature. At and below ${T}_{c}$, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.

World Scientific lecture notes in physicsNov 1, 19861 citation

Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses

Haim Sompolinsky, Annette Zippelius

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World Scientific lecture notes in physicsNov 1, 19861 citation

Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses

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Infinite-range-interactionM-component quantum spin glasses: Statics and dynamics in the large-Mlimit

T. K. Kopeć

The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a ${\mathit{T}}^{3}$ dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a ${\mathit{T}}^{3}$ dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.

Physical review. B, Condensed matterOct 1, 199414 citations

Short-range interactions: low-temperature properties

K. H. Fischer, John Hertz

The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).

Cambridge University Press eBooksMay 2, 19910 citations

The connection between statics and dynamics of spin glasses

Matthew Wittmann, A. P. Young

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arXiv:1504.07709Journal of Statistical Mechanics Theory and ExperimentJan 1, 20168 citations

\mathrm{XY}

{e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}

for

T

greater than

{T}_{c}

, the spin-glass transition temperature. At and below

{T}_{c}

, spin-spin correlations are observed to decay to their long-time limit as

{t}^{\ensuremath{-}\frac{1}{2}}

\mathrm{XY}

spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for

T

greater than

{T}_{c}

, the spin-glass transition temperature. At and below

{T}_{c}

, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.

{T}_{c}

. Approaching

{T}_{c}

from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at

{T}_{c}

the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below

{T}_{c}

. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below

{T}_{c}

. Consequently, the spin-glass order parameters are the time-persistent terms which appear in both the spin correlations and the local response. This is shown to invalidate the treatment of the spin-glass order parameters as purely static quantities. Instead, one has to specify the manner in which they relax in a finite system, along time scales which diverge in the thermodynamic limit. We show that the finite-time correlations decay algebraically with time as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ at all temperatures below

{T}_{c}

, with a temperature-dependent exponent $\ensuremath{\nu}$. Near

{T}_{c}

, $\ensuremath{\nu}$ is given (in the Ising case) as $\ensuremath{\nu}(T)\ensuremath{\sim}\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+\ensuremath{\sigma}{(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2}$. A tentative calculation of $\ensuremath{\nu}$ at

T=0

K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.

{T}_{c}

. Approaching

{T}_{c}

from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at

{T}_{c}

the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below

{T}_{c}

. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below

{T}_{c}

{T}_{c}

, with a temperature-dependent exponent $\ensuremath{\nu}$. Near

{T}_{c}

T=0

K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.

{\mathit{T}}^{3}

dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.