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Physical Review Letters

Fluctuations in Short-Range Spin-Glasses

Haim Sompolinsky, Annette Zippelius

Published Apr 25, 1983• 62 citations
Publisher

Abstract

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 $d=4$.

DOI
10.1103/physrevlett.50.1297

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Spatial fluctuations in spin-glasses are studied by an expansion to quadratic order around the dynamic mean-field theory. Below Tc{T}_{c}Tc​, a spatial decay of the "spin-glass susceptibility" [\ensuremathχ2(R\ensuremath→,\ensuremathω)]av{[{\ensuremath{\chi}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}, \ensuremath{\omega})]}_{\mathrm{av}}[\ensuremathχ2(R\ensuremath→​,\ensuremathω)]av​ is characterized at infinitesimal frequencies by a correlation length \ensuremathξ(\ensuremathω)\ensuremath{\xi}(\ensuremath{\omega})\ensuremathξ(\ensuremathω) which diverges as the inverse of the time-dependent Edwards-Anderson order parameter q(1\ensuremathω)q(\frac{1}{\ensuremath{\omega}})q(\ensuremathω1​) in the limit \ensuremathω\ensuremath→0\ensuremath{\omega}\ensuremath{\rightarrow}0\ensuremathω\ensuremath→0. The Fourier transform of equal-time correlations [〈S(0\ensuremath→)S(R\ensuremath→)〉2]av{[{〈S(\stackrel{\ensuremath{\rightarrow}}{0})S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉}^{2}]}_{\mathrm{av}}[〈S(0\ensuremath→​)S(R\ensuremath→​)〉2]av​ diverges as k\ensuremath−4{k}^{\ensuremath{-}4}k\ensuremath−4, signaling a breakdown of mean-field theory at low TTT below d=4d=4d=4.
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).
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.
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.
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.
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).
We propose a new picture of the Ising-spin-glass phase, based on an Ansatz for the scaling of low-lying large-scale-droplet excitations. We find behavior very different from the infinite-range model. The truncated spatial correlations decay as a power of distance, the ac nonlinear susceptibility diverges as a power of $\mathrm{ln}\ensuremath{\omega}$, and the magnetization noise power diverges as $\frac{1}{\ensuremath{\omega}}$ with logarithmic corrections. A magnetic field destroys the spin-glass phase so that there is no de Almeida-Thouless transition. Defect excitations should yield similar dynamic phenomena in vector spin-glasses.