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 ${|T-T_c|}^{-1\mathrm{}}$ whereas at $T_c$ the damping function vanishes as ${\omega }^{\frac{1}{2}}$. 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 $t^{-\nu }$ at all temperatures below $T_c$, with a temperature-dependent exponent $\nu$. Near $T_c$, $\nu$ is given (in the Ising case) as $\nu \mathrm{}\left(T\right)\mathrm{}\sim \frac{1}{2}-{\pi }^{-1\mathrm{}}\left(\frac{1-T}{T_c}\right)+\sigma {\left(\frac{1-T}{T_c}\right)}^2$. A tentative calculation of $\nu$ at $T=0\mathrm{}$ K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.

• Received 6 January 1982

DOI:https://doi.org/10.1103/PhysRevB.25.6860