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Physical review. B, Condensed matter

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

Haim Sompolinsky, Annette Zippelius

Published Jun 1, 1982• 479 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.25.6860

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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.

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Closed non-linear equations are derived through mode-coupling approximations for the correlation functions and susceptibilities for the Edwards-Anderson model of a classical Heisenberg system with nearest-neighbour random exchange couplings. These equations, which exhibit full dynamical stability, lead to a self-consistent treatment of spin-density propagation and spin-current relaxation. They are found to describe a transition from a paramagnetic to a spin glass phase, which is viewed as one where the dynamics changes from ergodic to a non-ergodic behaviour. Near the critical temperature T c the low-frequency correlation functions are obtained as dynamical scaling laws governed by a critical frequency omega c , which slows down proportionally to epsilon 2 , with epsilon =(T-T c )/T c . Approaching the critical point from the paramagnetic side the diffusivity vanishes linearly with epsilon . The susceptibility exhibits a symmetric cusp. The drastic changes of the spin dynamics at the transition point show up most clearly for the spin-current spectra. The numerical solution of the equations is discussed.

Journal of Physics C Solid State Physics•Nov 20, 1984•60 citations
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 Tc{T}_{c}Tc​. Approaching Tc{T}_{c}Tc​ from above, spin fluctuations slow down with a relaxation time proportional to ∣T\ensuremath−Tc∣\ensuremath−1{|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}∣T\ensuremath−Tc​∣\ensuremath−1 whereas at Tc{T}_{c}Tc​ the damping function vanishes as \ensuremathω12{\ensuremath{\omega}}^{\frac{1}{2}}\ensuremathω21​. We derive a criterion for dynamic stability below Tc{T}_{c}Tc​. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below Tc{T}_{c}Tc​. 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−\ensuremathν{t}^{\ensuremath{-}\ensuremath{\nu}}t\ensuremath−\ensuremathν at all temperatures below Tc{T}_{c}Tc​, with a temperature-dependent exponent \ensuremathν\ensuremath{\nu}\ensuremathν. Near Tc{T}_{c}Tc​, \ensuremathν\ensuremath{\nu}\ensuremathν is given (in the Ising case) as \ensuremathν(T)\ensuremath∼12\ensuremath−\ensuremathπ\ensuremath−1(1\ensuremath−TTc)+\ensuremathσ(1\ensuremath−TTc)2\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}\ensuremathν(T)\ensuremath∼21​\ensuremath−\ensuremathπ\ensuremath−1(Tc​1\ensuremath−T​)+\ensuremathσ(Tc​1\ensuremath−T​)2. A tentative calculation of \ensuremathν\ensuremath{\nu}\ensuremathν at T=0T=0T=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 Tc{T}_{c}Tc​. Approaching Tc{T}_{c}Tc​ from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at Tc{T}_{c}Tc​ the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below Tc{T}_{c}Tc​. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below Tc{T}_{c}Tc​. 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 Tc{T}_{c}Tc​, with a temperature-dependent exponent $\ensuremath{\nu}$. Near Tc{T}_{c}Tc​, $\ensuremath{\nu}$ 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=0T=0T=0 K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.
Exact averaged equations for spin relaxation in infinite-ranged spin-glasses are derived. It is shown that, in general, time-persistent terms in both correlation and response functions cannot be treated as purely static order parameters. A dynamic stability criterion for static solutions is derived. In the marginally stable solution dynamic correlations decay algebraically $\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\nu}}$ below Tc{T}_{c}Tc​, with an exponent which decreases continuously with temperature, $\ensuremath{\nu}(T)=\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+O({(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2})$, in disagreement with previous predictions ($\ensuremath{\nu}=\frac{1}{2}$).
We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for T→Tc+T\rightarrow T_{c}^{+}T→Tc+​. 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)O(1/N)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 LLL for the fields δQiαβ(t1,t2)δQ^{αβ}_{i}(t_{1},t_{2})δQiαβ​(t1​,t2​) up to the cubic order in the series expansion around the dynamic Mean-Field value Qαβ‾(t1,t2)\overline{{Q}^{αβ}}(t_{1},t_{2})Qαβ​(t1​,t2​). We find the more general expression for the time depending non-local fluctuations, the propagators [⟨δQiαβ(t1,t2)δQjαβ(t3,t4)⟩ξ]J[\langleδQ^{αβ}_{i}(t_{1},t_{2}) δQ^{αβ}_{j}(t_{3},t_{4})\rangle_ξ]_{J}[⟨δQiαβ​(t1​,t2​)δQjαβ​(t3​,t4​)⟩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).
We examine spin dynamics just above a spin-glass transition for two classical soft-spin models with random exchange interactions. The statics of the two models are identical; in mean-field theory we find an Edwards-Anderson transition of the usual sort. The dynamics are described by Langevin equations: in one case we use a purely dissipative equation of motion for a nonconserved scalar spin, while in the other we include a "mode-coupling" term to describe the precession of a vector spin in the field of its neighbors and impose conservation of the total spin. In the first case we find critical slowing down as the transition at Tg{T}_{g}Tg​ is approached from above (and ${t}^{\ensuremath{-}\frac{1}{2}}$ tails below Tg{T}_{g}Tg​), in agreement with the result of Ma and Rudnick for a different model. In the second case, we find a divergent spin-diffusion constant: $D\ensuremath{\propto}{(T\ensuremath{-}{T}_{g})}^{\ensuremath{-}\frac{1}{2}}$ (d>4d>4d>4) and $D\ensuremath{\propto}{(T\ensuremath{-}{T}_{g})}^{\frac{2}{d}\ensuremath{-}1}$ (2<d<42<d<42<d<4), in agreement with our previous calculation for a mode-coupling version of the Ma-Rudnick model.
The dynamical properties of a simple model of a spin glass are studied. The model has Gaussian random exchange interactions and Langevin dynamics. Following the ideas of Ma and Rudnick (1978) and of De Dominicis (1978), the spin glass phase is characterised by the presence of a finite limit of the spin autocorrelation function as t to infinity , i.e. by a delta-function part of it Fourier transform C( omega ). With a finite system in mind, this delta function is taken to have a small finite width. Then the fluctuation-dissipation theorem is imposed (before the infinite volume limit), forcing a corresponding low-frequency anomaly on the dynamical susceptibility. Solving self-consistently in mean-field theory for the order parameter, one then finds that the Almeida-Thouless instability of conventional theories is suppressed for all times shorter than the inverse width of the delta function in C( omega ). This theory is constructed for exactly zero external field; problems about the field dependence and possible relevance to experiment are discussed.
Closed non-linear equations are derived through mode-coupling approximations for the correlation functions and susceptibilities for the Edwards-Anderson model of a classical Heisenberg system with nearest-neighbour random exchange couplings. These equations, which exhibit full dynamical stability, lead to a self-consistent treatment of spin-density propagation and spin-current relaxation. They are found to describe a transition from a paramagnetic to a spin glass phase, which is viewed as one where the dynamics changes from ergodic to a non-ergodic behaviour. Near the critical temperature T c the low-frequency correlation functions are obtained as dynamical scaling laws governed by a critical frequency omega c , which slows down proportionally to epsilon 2 , with epsilon =(T-T c )/T c . Approaching the critical point from the paramagnetic side the diffusivity vanishes linearly with epsilon . The susceptibility exhibits a symmetric cusp. The drastic changes of the spin dynamics at the transition point show up most clearly for the spin-current spectra. The numerical solution of the equations is discussed.