We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for
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) 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
L for the fields
δQiαβ(t1,t2) up to the cubic order in the series expansion around the dynamic Mean-Field value
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, 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).
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.
A Landau-Ginzburg phenomenological free energy for the Edwards and Anderson spin-glass model when there is competition between spin-glass and ferromagnetic ordering is developed. This free energy obtained with the use of the $n\ensuremath{\rightarrow}0$ replication procedure is analyzed using mean-field theory and the $\ensuremath{\epsilon}$ expansion. Critical exponents for the ferromagnetic-spin-glass multicritical point are calculated in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions. For Ising systems, $\ensuremath{\nu}=\frac{1}{2}+(\frac{2}{3})\ensuremath{\epsilon}$ and $\ensuremath{\varphi}=1+(\frac{1}{2})\ensuremath{\epsilon}$. For
XY and Heisenberg systems, these exponents are complex. This result is not fully understood. The Harris-Plischke-Zuckermann model for amorphous magnetism is shown to have an Ising-like spin-glass fixed point in high enough dimension.
The critical dynamics of the time-dependent Ginzburg-Landau model for a system with quenched random impurities and nonconserved order parameter is studied in the framework of the $\ensuremath{\epsilon}$ expansion. In contrast to the situation in pure systems, the dynamic critical exponent
z deviates from its conventional value at first order in $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The impurities cause an enhancement of the shape function ${f}_{x}(\ensuremath{\nu})$ at small frequencies $\ensuremath{\nu}$; ${f}_{x}(\ensuremath{\nu}=0)$ diverges as $T\ensuremath{\rightarrow}{T}_{c}$. Below
Tc the equation of state, static susceptibility $\ensuremath{\chi}$, and dynamic response function
G, are studied. A new, purely static correlated function,
C(s), whose existence is unique to the random system is introduced. The coexistence curve singularities of
C(s), $\ensuremath{\chi}$, and
G in systems with continuously broken symmetry are explored. The connection of the quenched-impurity model with "model
C" of Halperin, Hohenberg, and Ma is discussed.
We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with nonconserved n-component order parameter (model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations $g(|r\ensuremath{-}{r}^{\ensuremath{'}}|)\ensuremath{\sim}|r\ensuremath{-}{r}^{\ensuremath{'}}{|}^{\ensuremath{-}a},$ $a=4\ensuremath{-}\ensuremath{\delta},$ previously introduced by Weinrib and Halperin. The parameter $\ensuremath{\delta}\ensuremath{\sim}\ensuremath{\epsilon},$ where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the usual small parameter within the Wilson-Fisher renormalization group theory. Using a double expansion in $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ we calculate the critical exponent
z up to second order on the small parameters. We show that quenched impurities of this kind affect the critical dynamics already in first order of $\ensuremath{\epsilon}$ and $\ensuremath{\delta},$ leading to a relevant correction for the mean field value of the exponent
z.