Using the zero-temperature Metropolis dynamics, the time decay of the remanent magnetization in the +/-J Edward-Anderson spin glass model with a uniform random distribution of ferromagnetic and antiferromagnetic interactions has been investigated. Starting from the saturation, the magnetization per spin m reveals a slow decrease with time, which can be approximated by a power law: m(t)=m(infinity)+(t/a(0))(a(1)), a(1)<0. Moreover, its relaxation does not lead it into one of the ground states, and therefore the system is trapped in metastable isoenergetic microstates remaining magnetized. Such behavior is discussed in terms of a random walk that the system performs on its available configuration space.
The remanent magnetization
Mr and the excess energy \ensuremath{\Delta}E are studied for the Sherrington-Kirpatrick model of spin glasses using Monte Carlo simulations. At zero temperature T the system relaxes to a stationary nonequilibrium state with nonzero
Mr and \ensuremath{\Delta}E; both quantities are self-averaging. For 0
Tc the simulations indicate that
Mr and \ensuremath{\Delta}E decay to zero as a power law in time.
We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.
A quench of a d-dimensional spin system from a random initial configuration, {
Si(0)}, to a critical point is considered. The decay with time t of the autocorrelation with the initial condition is
q0(t)==〈
Si(0)\ensuremath{\cdot}
Si(t)〉\ensuremath{\sim}${t}_{\mathit{c}}^{\ensuremath{-}\ensuremath{\lambda}}$/z, where z is the usual dynamic critical exponent. Naively, ${\ensuremath{\lambda}}_{c}$=d, but I find ${\ensuremath{\lambda}}_{c}$simulations of pure Ising models in d=2, 3 and the \ifmmode\pm\else\textpm\fi{}J Ising spin glass in d=3. This suggests that ${\ensuremath{\lambda}}_{c}$ is a new critical exponent for nonequilibrium dynamics. For a spin glass the decay of
q0(t) is the same as that of the remanent magnetization; the exponent ${\ensuremath{\lambda}}_{c}$/z observed in the spin-glass simulation is in good agreement with a recent experimental measurement by Granberg et al.
The low temperature dynamics of the three-dimensional Ising spin-glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t, t w ) shows a crossover at the waiting (or aging) time t w from algebraic quasiequilibrium decay for times t<<t w to another, faster algebraic decay for t>>t w with an exponent similar to one for the remanent magnetization.
The low temperature dynamics of the three dimensional Ising spin glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t,tw) shows a crossover at the waiting (or aging) time tw from algebraic quasi-equilibrium decay for times t≪tw to another, faster algebraic decay for t≫tw with an exponent similar to one for the remanent magnetization.
We study the time-dependent properties of an Ising chain with random bonds Gaussianly distributed. We obtain some exact as well as Monte Carlo (MC) results. Remanence, as well as a seemingly logarithmic long-time decay of the magnetization and of the energy towards their equilibrium values is observed at low temperatures by means of MC simulation. The remanent values of the magnetization (
31, starting with all spins up) and of the energy are derived. A simple and explicit physical picture of the mechanism behind remanence and the logarithmic relaxation emerges. The equilibrium value of $q(t)={〈〈{\ensuremath{\sigma}}_{i}(0){\ensuremath{\sigma}}_{i}(t)〉〉}_{J}$ is obtained via the MC technique; it also seems to relax logarithmically for low temperatures. In contrast with the two-and three-dimensional cases, it is shown how any MC calculation can start immediately from an "equilibrium state" in this model, a very convenient feature for very-low-temperature MC computations. We show that if
H is switched on at
t=0, then ${[\frac{\ensuremath{\partial}m(H,t)}{\ensuremath{\partial}H}]}_{H=0}=(\frac{1}{\mathrm{kT}})[1\ensuremath{-}q(t)]$ holds exactly in any number of dimensions, where
m is the magnetization per spin. A time-dependent susceptibility $\ensuremath{\chi}(H,t)$ is defined and shown to vanish (as $H\ensuremath{\rightarrow}0$) for low enough temperatures and finite
t. The MC results for $\ensuremath{\chi}$ are in accord with this result and, if graphed versus
T, show a hump at a time-dependent temperature. Finally, we compare the exact equilibrium specific heat,
C, for this model with the results obtained by MC simulation of calorimetric measurements. Thus, a simple explicit case is exhibited of the difficulties which may arise in measurements of
C in spin-glasses due to long-time effects.