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. Approaching
Tc from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at
Tc the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below
Tc. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below
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, with a temperature-dependent exponent $\ensuremath{\nu}$. Near
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=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. Approaching
Tc from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at
Tc the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below
Tc. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below
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, with a temperature-dependent exponent $\ensuremath{\nu}$. Near
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=0 K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.
The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.
We discuss the dynamics of the infinite-range Sherrington-Kirkpatrick spin-glass model for which relaxation times diverge when
N, the number of spins, tends to infinity. Calculations on a large but finite system are very difficult, so we mimic a large finite system in equilibrium by working with $N=\ensuremath{\infty}$ and imposing, by hand, a canonical distribution at an initial time. For short times, where no barrier hopping has occurred, we find that the Edwards-Anderson order parameter,
qEA, is identical to that obtained from an analysis of the mean-field equations of Thouless, Anderson, and Palmer and, with further assumptions, gives
q(x=1) in Parisi's theory, in agreement with earlier work. For times longer than the longest relaxation time (of the finite system), true equilibrium is reached and our theory agrees with previous statistical-mechanics calculations using the replica trick. There is no violation of the fluctuation-dissipation theorem.
We discuss the derivation of canonical correlation functions from the dynamic approach in long-range spin-glasses. We argue that such derivations are possible using the saddle-point approximation even in the presence of ergodicity breaking, within the following scenario: a ) the dynamical saddle-point solution is not unique at infinite times-scales; b ) a single dynamic solution does not yield the measurable correlation functions ( e.g. as observed at very long times in a computer simulation) but these have to be extracted from a sum over all such dynamic saddle-point processes. We propose a conjecture on how the different infinite-time solutions to the mean-field dynamical equations are related by a certain group of transformations. We then show that if all relevant saddle-points are obtained from Sompolinsky's solution through these transformations, the resulting P ( q ) is identical to that obtained with the Parisi scheme in replica space.
It is proposed that the spin-glass phase is described by two order parameters
q(x)=[〈Si(0)Si(tx)〉]av 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,
tx, which are parametrized in a decreasing order by $x\ensuremath{\in}[0,1]$. The function
q decays from a finite value at
t1 to zero at
t0, 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.