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Physical Review Letters

Time-Dependent Order Parameters in Spin-Glasses

Haim Sompolinsky

Published Sep 28, 1981• 380 citations
Publisher

Abstract

It is proposed that the spin-glass phase is described by two order parameters $q(x)={[〈{S}_{i}(0){S}_{i}({t}_{x})〉]}_{\mathrm{a}\mathrm{v}}$ and $\ensuremath{\chi}(x)=\ensuremath{\int}{0}^{{t}_{x}}\ensuremath{\chi}(t)\mathrm{dt}$, that measure the relaxations of the average autocorrelation and susceptibility along macroscopic time scales, ${t}_{x}$, which are parametrized in a decreasing order by $x\ensuremath{\in}[0,1]$. The function $q$ decays from a finite value at ${t}_{1}$ to zero at ${t}_{0}$, while $\ensuremath{\chi}$ increases from $\ensuremath{\chi}(1)$ to a value $\ensuremath{\chi}(0)$ which is independent of temperature in the mean-field limit. The equilibrium results agree with Parisi's replica solution.

DOI
10.1103/physrevlett.47.935

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

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Physical review. B, Condensed matter•Jun 1, 1982•479 citations
It is proposed that the spin-glass phase is described by two order parameters q(x)=[〈Si(0)Si(tx)〉]avq(x)={[〈{S}_{i}(0){S}_{i}({t}_{x})〉]}_{\mathrm{a}\mathrm{v}}q(x)=[〈Si​(0)Si​(tx​)〉]av​ and \ensuremathχ(x)=\ensuremath∫0tx\ensuremathχ(t)dt\ensuremath{\chi}(x)=\ensuremath{\int}{0}^{{t}_{x}}\ensuremath{\chi}(t)\mathrm{dt}\ensuremathχ(x)=\ensuremath∫0tx​\ensuremathχ(t)dt, that measure the relaxations of the average autocorrelation and susceptibility along macroscopic time scales, tx{t}_{x}tx​, which are parametrized in a decreasing order by x\ensuremath∈[0,1]x\ensuremath{\in}[0,1]x\ensuremath∈[0,1]. The function qqq decays from a finite value at t1{t}_{1}t1​ to zero at t0{t}_{0}t0​, while \ensuremathχ\ensuremath{\chi}\ensuremathχ increases from \ensuremathχ(1)\ensuremath{\chi}(1)\ensuremathχ(1) to a value \ensuremathχ(0)\ensuremath{\chi}(0)\ensuremathχ(0) which is independent of temperature in the mean-field limit. The equilibrium results agree with Parisi's replica solution.
A simple approximation procedure is introduced which quantitatively accounts for the low-temperature logarithmic decay in time of the autocorrelation function for the Edwards-Anderson model of an Ising spin glass. The relationship between the autocorrelation function and the susceptibility is used to derive the dependence on observation time of the susceptibility. The order parameter q(t) is calculated and is in good agreement with the Monte Carlo simulations on two- and three-dimensional systems with Gaussian bond distributions, which show that q(t) tends to zero for long times t.
The time-dependent order parameter q(t 2 -t 1 ) is re-examined as a function of the time difference t 2 -t 1 and number of spins N, for the infinite-range spin glass. In the statistical mechanics limit, t 2 -t 1 to infinity before N to infinity , the authors find q(t 2 -t 1 )=q= integral q(x)dx, where q(x) is the order parameter function of Parisi (1979, 1980, 1983), provided the system has a canonical (Boltzmann) distribution at the earlier time t 1 . In the opposite limit, N to infinity before t 2 -t 1 to infinity , they obtain q(t 2 -t 1 )=q(x=1). If the initial distribution differs slightly from a canonical one they obtain q(t 2 , t 1 )=q(x=0) (i.e. the statistical mechanics limit) and the Sompolinsky solution (1981) appears to be correct under these (non-equilibrium) conditions.
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 review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.
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.