Here is presented a dynamic theory of spin glasses based on the extended mean-field theory of Thouless, Anderson, and Palmer. The kinetic equations for the magnetization and the spin correlation function are derived, from which dynamical linear and nonlinear susceptibilities are calculated and their low-frequency behavior is mainly investigated. The spin self-correlation functions have different relaxation times near above and below T c –the former is proportional to | T - T c | -2 , while the latter to | T - T c | -4 –which shows the asymmetric critical slowing down. A qualitative comparison of the present results with experimental data is presented.
Here is presented a dynamic theory of spin glasses based on the extended mean-field theory of Thouless, Anderson, and Palmer. The kinetic equations for the magnetization and the spin correlation function are derived, from which dynamical linear and nonlinear susceptibilities are calculated and their low-frequency behavior is mainly investigated. The spin self-correlation functions have different relaxation times near above and below T c –the former is proportional to | T - T c | -2 , while the latter to | T - T c | -4 –which shows the asymmetric critical slowing down. A qualitative comparison of the present results with experimental data is presented.
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.
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.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for T greater than Tc, the spin-glass transition temperature. At and below Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for T greater than Tc, the spin-glass transition temperature. At and below Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A general phenomenological theory of spin-glasses is presented, which predicts the weak singularities (like cusps) of susceptibilities and specific heat near the transition point TSG and also predicts generally the divergence of the second derivative of the nonlinear susceptibility with respect to a magnetic field in zero field at TSG. A similar second derivative of specific heat is also shown to diverge at TSG except for the molecular field theory. Scaling equations of state for magnetization m and spin-glass order-parameter q are derived, which yield new scaling relations of critical exponents for m(T), q(T), χ0(T), C0(T) and the corresponding nonlinear quantities. These general results explain qualitatively very well the magnetic field dependence of magnetization and specific heat observed experimentally. A non-uniform phenomenological Hamiltonian is also proposed for the renormalization group approach, which yields the critical dimensionality dc = 6, as usual. Intermediate random statistics and real replica method are proposed. There is also presented an exactly soluble model showing a spin-glass like phase transition.