A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.
We have carried out extensive Monte Carlo simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic classical Heisenberg models with periodic boundary conditions. The equilibrium relaxation time \ensuremath{\tau} is determined from the long-time exponential decay of the time-displaced correlation function of the magnetization. Using finite-size scaling and corrections to scaling we find that at the critical point
Tc=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}
Lz with z=1.96(6). In the paramagnetic region we show that the relaxation time is represented by a universal scaling function \ensuremath{\tau}${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{z}}$ vs (T/
Tc-1)${\mathit{L}}^{1/\ensuremath{\nu}}$ with z=1.96(6). This estimate is consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving dipolar interactions. We also describe how a simple over-relaxation algorithm of Creutz and of Brown and Woch can be incorporated into the Metropolis sampling method for the classical Heisenberg model, thereby significantly reducing the critical slowing down; the resultant correlation time varies as \ensuremath{\tau}\ensuremath{\sim}(T/
Tc-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.
We investigate nonequilibrium relaxations of Ising models at the critical point by using a cluster update. While preceding studies imply that nonequilibrium cluster-flip dynamics at the critical point are universally described by the stretched-exponential function, we find that the dynamics changes from the stretched exponential to the power function as the dimensionality is increased: The two-, three-, four-, and infinite-dimensional Ising models are numerically studied, and the four- and infinite-dimensional Ising models exhibit the power-law relaxation. We also show that the finite-size scaling analysis using the normalized correlation length is markedly effective for the analysis of relaxational processes rather than the direct use of the Monte Carlo step.
We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization
M,MSDM, as a function of time, as well as on the autocorrelation function of
M. These two functions are distinct but closely related. We find that
MSDM features a first crossover at time ${\ensuremath{\tau}}_{1}\ensuremath{\sim}{L}^{{z}_{1}}$, from ordinary diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}t$, to anomalous diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$. Purely on numerical grounds, we obtain the values
z1=0.45(5) and $\ensuremath{\alpha}=0.752(5)$ for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time ${\ensuremath{\tau}}_{2}\ensuremath{\sim}{L}^{{z}_{2}}$. Here,
MSDM saturates to its late-time value $\ensuremath{\sim}{L}^{2+\ensuremath{\gamma}/\ensuremath{\nu}}$, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value
z2=2.1665(12), earlier reported as the single dynamic exponent. Continuity of
MSDM requires that $\ensuremath{\alpha}({z}_{2}\ensuremath{-}{z}_{1})=\ensuremath{\gamma}/\ensuremath{\nu}\ensuremath{-}{z}_{1}$. We speculate that
z1=1/2 and $\ensuremath{\alpha}=3/4$, values that indeed lead to the expected
z2=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates ${z}_{1}=1.35(2), \ensuremath{\alpha}=0.90(2)$, and
z2=2.032(3). While
z2 has attracted significant attention in the literature, we argue that for all practical purposes
z1 is more important, as it determines the number of statistically independent measurements during a long simulation.
The energy relaxation in the two-dimensional Ising model, with temperature greater than
Tc and in a spin system having a 5120\ifmmode\times\else\texttimes\fi{}5120 lattice, is studied using supercomputer-conducted Monte Carlo simulation. The critical exponents of the linear and nonlinear relaxation times of the internal energy are respectively estimated to be ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=2.064\ifmmode\pm\else\textpm\fi{}0.026 and ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathrm{nl})}$=1.132\ifmmode\pm\else\textpm\fi{}0.080, being consistent with the Racz scaling law ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$-${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathrm{nl})}$=1-\ensuremath{\alpha}, where \ensuremath{\alpha} is the critical exponent of the specific heat. Based on the author's previous result in which the critical exponent of the linear relaxation for magnetization was determined as ${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$=2.076\ifmmode\pm\else\textpm\fi{}0.005, it is strongly suggested that ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$.
A model has been developed that associates a hierarchical criterion to the usual renormalization of the distances near a phase transition. At times t smaller than τα∝(1−Tc/T)−zν, which diverges at Tc, it is found that the relaxation develops at a scale s(t), which is a power law of the time elapsed: s ∝ tT/zTc. If one sits in the ergodic regime T≳Tc, the system abruptly reaches an equilibrium at s(τα)=ξ(T)=τα1/z. If T<Tc, by contrast, it deviates to logarithmic behavior. The α relaxation is well approximated by a stretched exponential exp(−t/τα)β(T), which traduces the effect of underlying lognormal statistics. The exponent β(T)∝[1−b ln(1−Tc/T)]−1/2 would cancel at Tc if it was possible to wait for the sample to equilibrate at this temperature. The model has the necessary ingredients to describe effects that have been reported in both the ‘‘activated’’ nonergodic regime and in the critical regime of spin glasses, and of other correlated systems like orientational glasses, glass-forming liquids, and vortex glasses. A possible extension of the model is discussed to describe the ‘‘macroscopic quantum effects.’’