arXiv:cond-mat/9501027 · Journal de Physique I

Dynamic Fluctuations in a Short-range Spin Glass Model

Paola Ranieri

Published Jun 1, 1996• 4 citations

Abstract

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for $T\rightarrow T_{c}^{+}$. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order $O(1/N)$ and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian $L$ for the fields $δQ^{αβ}_{i}(t_{1},t_{2})$ up to the cubic order in the series expansion around the dynamic Mean-Field value $\overline{{Q}^{αβ}}(t_{1},t_{2})$. We find the more general expression for the time depending non-local fluctuations, the propagators $[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}$, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices).

DOI: 10.1051/jp1:1996243

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

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arXiv:cond-mat/9501027 · Journal de Physique I

Dynamic Fluctuations in a Short-range Spin Glass Model

Paola Ranieri

Published Jun 1, 1996• 4 citations

arXiv HTML PDF

Abstract

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for

T\rightarrow T_{c}^{+}

. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order

O(1/N)

and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian

L

for the fields

δQ^{αβ}_{i}(t_{1},t_{2})

up to the cubic order in the series expansion around the dynamic Mean-Field value

\overline{{Q}^{αβ}}(t_{1},t_{2})

. We find the more general expression for the time depending non-local fluctuations, the propagators

[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}

, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices).

DOI: 10.1051/jp1:1996243

Related papers

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.

Physical review. B, Condensed matterJun 1, 1982479 citations

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

Haim Sompolinsky, Annette Zippelius

World Scientific lecture notes in physicsNov 1, 19861 citation

Fluctuations in Short-Range Spin-Glasses

Haim Sompolinsky, Annette Zippelius

Spatial fluctuations in spin-glasses are studied by an expansion to quadratic order around the dynamic mean-field theory. Below ${T}_{c}$ , a spatial decay of the "spin-glass susceptibility" ${[{\ensuremath{\chi}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}, \ensuremath{\omega})]}_{\mathrm{av}}$ is characterized at infinitesimal frequencies by a correlation length $\ensuremath{\xi}(\ensuremath{\omega})$ which diverges as the inverse of the time-dependent Edwards-Anderson order parameter $q(\frac{1}{\ensuremath{\omega}})$ in the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$. The Fourier transform of equal-time correlations ${[{〈S(\stackrel{\ensuremath{\rightarrow}}{0})S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉}^{2}]}_{\mathrm{av}}$ diverges as ${k}^{\ensuremath{-}4}$, signaling a breakdown of mean-field theory at low $T$ below $d=4$ .Spatial fluctuations in spin-glasses are studied by an expansion to quadratic order around the dynamic mean-field theory. Below ${T}_{c}$, a spatial decay of the "spin-glass susceptibility" ${[{\ensuremath{\chi}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}, \ensuremath{\omega})]}_{\mathrm{av}}$ is characterized at infinitesimal frequencies by a correlation length $\ensuremath{\xi}(\ensuremath{\omega})$ which diverges as the inverse of the time-dependent Edwards-Anderson order parameter $q(\frac{1}{\ensuremath{\omega}})$ in the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$. The Fourier transform of equal-time correlations ${[{〈S(\stackrel{\ensuremath{\rightarrow}}{0})S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉}^{2}]}_{\mathrm{av}}$ diverges as ${k}^{\ensuremath{-}4}$, signaling a breakdown of mean-field theory at low $T$ below $d=4$.

Physical Review LettersApr 25, 198362 citations

Quantum spherical spin glass with random short-range interactions

Pedro Castro Menezes, Alba Theumann

In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part ${W}_{\text{MF}}$ coincides with previous results. The short-range part ${W}_{\text{SR}}$ describes a phase transition in a ${Q}^{3}$ -field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as ${d}_{c}=5$ at very low temperatures due to the dimensionality shift ${D}_{c}={d}_{c}+1=6$ . We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part ${W}_{\text{MF}}$ coincides with previous results. The short-range part ${W}_{\text{SR}}$ describes a phase transition in a ${Q}^{3}$-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as ${d}_{c}=5$ at very low temperatures due to the dimensionality shift ${D}_{c}={d}_{c}+1=6$. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.

arXiv:0806.0028Physical Review BAug 25, 20085 citations

Quantum spin glass with long-range random interactions

Amit Dutta

A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance ${r}_{\mathrm{ij}}$ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance ${r}_{\mathrm{ij}}$ apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.

Physical review. B, Condensed matterJun 10, 20029 citations

Static and dynamic magnetic response of spin-glass models with short-range interactions

Wolfgang Kinzel, Kurt Binder

The static and dynamic magnetic response of the two-dimensional Edwards-Anderson model with a nearest-neighbor Gaussian exchange distribution is investigated by Monte Carlo simulation. A plateau in the equilibrium (slowly field cooled) susceptibility $\frac{{M}_{\mathrm{eq}}}{H}$ is found, which diverges for small fields with a power law $\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}$, $\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5$. The boundary of the plateau ${H}_{c}^{\mathrm{eq}}(T)$ tends to zero temperature as ${H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}$, consistent with a scaling description appropriate for a static phase transition at $T=0$ , and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures ${T}_{f}$ , in striking similarity to experimental data, but ${T}_{f}=0$ is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from $\frac{{M}_{\mathrm{eq}}}{H}$ below a certain critical field ${H}_{c}(t)$ , $t$ being the time scale over which the field is applied. In the $H\ensuremath{-}T$ plane ${H}_{c}(t)$ extrapolates to a time-dependent freezing temperature ${T}_{c}(t)$ , and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.The static and dynamic magnetic response of the two-dimensional Edwards-Anderson model with a nearest-neighbor Gaussian exchange distribution is investigated by Monte Carlo simulation. A plateau in the equilibrium (slowly field cooled) susceptibility $\frac{{M}_{\mathrm{eq}}}{H}$ is found, which diverges for small fields with a power law $\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}$, $\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5$. The boundary of the plateau ${H}_{c}^{\mathrm{eq}}(T)$ tends to zero temperature as ${H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}$, consistent with a scaling description appropriate for a static phase transition at $T=0$, and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures ${T}_{f}$, in striking similarity to experimental data, but ${T}_{f}=0$ is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from $\frac{{M}_{\mathrm{eq}}}{H}$ below a certain critical field ${H}_{c}(t)$, $t$ being the time scale over which the field is applied. In the $H\ensuremath{-}T$ plane ${H}_{c}(t)$ extrapolates to a time-dependent freezing temperature ${T}_{c}(t)$, and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.

Physical review. B, Condensed matterFeb 1, 198452 citations

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for

T\rightarrow T_{c}^{+}

. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order

O(1/N)

and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian

L

for the fields

δQ^{αβ}_{i}(t_{1},t_{2})

up to the cubic order in the series expansion around the dynamic Mean-Field value

\overline{{Q}^{αβ}}(t_{1},t_{2})

. We find the more general expression for the time depending non-local fluctuations, the propagators

[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}

{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.

Spatial fluctuations in spin-glasses are studied by an expansion to quadratic order around the dynamic mean-field theory. Below

{T}_{c}

, a spatial decay of the "spin-glass susceptibility" ${[{\ensuremath{\chi}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}, \ensuremath{\omega})]}_{\mathrm{av}}$ is characterized at infinitesimal frequencies by a correlation length $\ensuremath{\xi}(\ensuremath{\omega})$ which diverges as the inverse of the time-dependent Edwards-Anderson order parameter $q(\frac{1}{\ensuremath{\omega}})$ in the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$. The Fourier transform of equal-time correlations ${[{〈S(\stackrel{\ensuremath{\rightarrow}}{0})S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉}^{2}]}_{\mathrm{av}}$ diverges as ${k}^{\ensuremath{-}4}$, signaling a breakdown of mean-field theory at low

T

below

d=4

{W}_{\text{MF}}

coincides with previous results. The short-range part

{W}_{\text{SR}}

describes a phase transition in a

{Q}^{3}

-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as

{d}_{c}=5

at very low temperatures due to the dimensionality shift

{D}_{c}={d}_{c}+1=6

. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.

A model of the quantum (transverse) Ising spin glass with a long-range random interaction is proposed and studied here. In this model the random interaction between two spins at a distance

{r}_{\mathrm{ij}}

apart falls algebraically as ${1/r}_{\mathrm{ij}}^{(d+\ensuremath{\sigma})/2}.$ Here, apart from the strength of the quantum fluctuations, the interaction range $\ensuremath{\sigma}$ is also tunable. We have studied the ground-state properties of the model, extending the ``Droplet model'' of the short-range quantum spin glass. The model is also studied using a modified form of the effective Landau action describing the transition of the short-range quantum spin glass. The important features which are due to the long-range interaction are clearly mentioned. Field theoretical renormalization group calculations fail to locate any stable weak-coupling fixed point in the non-mean-field region. The simplest conjecture is that beyond the mean-field region, the critical behavior is governed by the infinite randomness fixed point. We extend the phenomenological scaling relation for the short-range quantum spin glass (based on the assumption of the existence of a dangerously irrelevant operator) to the present long-range interacting case. Most of the possible interesting aspects associated with the quantum transition in the present model are elaborately discussed.

\frac{{M}_{\mathrm{eq}}}{H}

is found, which diverges for small fields with a power law $\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}$, $\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5$. The boundary of the plateau

{H}_{c}^{\mathrm{eq}}(T)

tends to zero temperature as ${H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}$, consistent with a scaling description appropriate for a static phase transition at

T=0

, and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures

{T}_{f}

, in striking similarity to experimental data, but

{T}_{f}=0

is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from

\frac{{M}_{\mathrm{eq}}}{H}

below a certain critical field

{H}_{c}(t)

t

being the time scale over which the field is applied. In the $H\ensuremath{-}T$ plane

{H}_{c}(t)

extrapolates to a time-dependent freezing temperature

{T}_{c}(t)

, and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.