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.
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.
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.
We examine spin dynamics just above a spin-glass transition for two classical soft-spin models with random exchange interactions. The statics of the two models are identical; in mean-field theory we find an Edwards-Anderson transition of the usual sort. The dynamics are described by Langevin equations: in one case we use a purely dissipative equation of motion for a nonconserved scalar spin, while in the other we include a "mode-coupling" term to describe the precession of a vector spin in the field of its neighbors and impose conservation of the total spin. In the first case we find critical slowing down as the transition at Tg is approached from above (and ${t}^{\ensuremath{-}\frac{1}{2}}$ tails below Tg), in agreement with the result of Ma and Rudnick for a different model. In the second case, we find a divergent spin-diffusion constant: $D\ensuremath{\propto}{(T\ensuremath{-}{T}_{g})}^{\ensuremath{-}\frac{1}{2}}$ (d>4) and $D\ensuremath{\propto}{(T\ensuremath{-}{T}_{g})}^{\frac{2}{d}\ensuremath{-}1}$ (2<d<4), in agreement with our previous calculation for a mode-coupling version of the Ma-Rudnick model.
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.
Slow cooling dynamics of the Ising p-spin interaction spin-glass model
We have studied dynamical behaviour of the infinite-range Ising spin glass model with p-spin interaction above and below the transition into the non-ergodic phase. The transition is continuous at sufficiently high external magnetic field. The dynamic critical exponent of the power-law decay of the autocorrelation function at the transition point is shown to decrease smoothly to zero as the field approaches the ``tricritical'' point from above; at lower fields the transition is discontinuous. The {\it slow cooling} approach is used to study the nonergodic behavior below the transition at zero external field. It is shown that the anomalous response function Δ(t,t′) contains δ-function as well as regular contributions at {\it any} temperature below the phase transition. No evidence of the second phase transition (known to exist within the static replica solution of the same model) is found. At lower enough temperatures the {\it slow cooling} solution approaches the one known for the standard SK model.
This thesis aims at a description of the equilibrium dynamics of quantum spin glass systems. To this end a generic fermionic SU(2), spin 1/2 spin glass model with infinite-range interactions is defined in the first part. The model is treated in the framework of imaginary-time Grassmann field theory along with the replica formalism. A dynamical two-step decoupling procedure, which retains the full time dependence of the (replica-symmetric) saddle point, is presented. As a main result, a set of highly coupled self-consistency equations for the spin-spin correlations can be formulated. Beyond the so-called spin-static approximation two complementary systematic approximation schemes are developed in order to render the occurring integration problem feasible. One of these methods restricts the quantum-spin dynamics to a manageable number of bosonic Matsubara frequencies. A sequence of improved approximants to some quantity can be obtained by gradually extending the set of employed discrete frequencies. Extrapolation of such a sequence yields an estimate of the full dynamical solution. The other method is based on a perturbative expansion of the self-consistency equations in terms of the dynamical correlations. In the second part these techniques are applied to the isotropic Heisenberg spin glass both on the Fock space (HSGF) and, exploiting the Popov-Fedotov trick, on the spin space (HSGS). The critical temperatures of the paramagnet to spin glass phase transitions are determined accurately. Compared to the spin-static results, the dynamics causes slight increases of T_c by about 3% and 2%, respectively. For the HSGS the specific heat C(T) is investigated in the paramagnetic phase and, by way of a perturbative method, below but close to T_c. The exact C(T)-curve is shown to exhibit a pronounced non-analyticity at T_c and, contradictory to recent reports by other authors, there is no indication of maximum above T_c. In the last part of this thesis the spin glass model is augmented with a nearest-neighbor hopping term on an infinite-dimensional cubic lattice. An extended self-consistency structure can be derived by combining the decoupling procedure with the dynamical CPA method. For the itinerant Ising spin glass numerous solutions within the spin-static approximation are presented both at finite and zero temperature. Systematic dynamical corrections to the spin-static phase diagram in the plane of temperature and hopping strength are calculated, and the location of the quantum critical point is determined.