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: 10.1142/9789812799371_0011

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

Physical review. B, Condensed matterJun 1, 19781,007 citations

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

Haim Sompolinsky, Annette Zippelius

Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below ${T}_{c}$ . 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.Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below ${T}_{c}$. 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.

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

Physical review. B, Condensed matterJun 1, 1982479 citations

Infinite-range Ising spin glass with a transverse field under the static approximation

Dohyun Kim, Jong-Jean Kim

In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of ${C}_{H}/T$ near the glass transition temperature ${T}_{g},$ associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of ${C}_{H}/T$ near the glass transition temperature ${T}_{g},$ associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.

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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

The connection between statics and dynamics of spin glasses

Matthew Wittmann, A. P. Young

Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.

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.

{C}_{H}/T

near the glass transition temperature

{T}_{g},

associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.

{\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.