In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.
We simulate the critical relaxation process of the two-dimensional Ising model with the initial state both completely disordered or completely ordered. Results of a method to measure both the dynamic and static critical exponents are reported, based on the finite-size scaling for the dynamics at the early time. From the time-dependent Binder cumulant, the dynamical exponent z is extracted independently, while the static exponents \ensuremath{\beta}/\ensuremath{\nu} and \ensuremath{\nu} are obtained from the time evolution of the magnetization and its higher moments. \textcopyright{} 1996 The American Physical Society.
We obtain the exact critical relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi}), where \ensuremath{\xi} is the bulk correlation length, for the Glauber kinetic Ising model of spins on a one-dimensional lattice of finite length L for both periodic and free boundary conditions (BC's). We show that, independent of the BC's, the dynamic critical exponent has the well-known value z=2, and we comment on a recent claim that z=1 for this model. The ratio ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi})/${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\xi}), in the double limit L,\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty} for fixed x=L/\ensuremath{\xi}, approaches a limiting functional form, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(L/\ensuremath{\xi}), the finite-size scaling function. For free BC's we derive the exact scaling function ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\omega}(x)/x
)2${]}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\omega}(x) is the smallest root of the transcendental equation \ensuremath{\omega}tan(\ensuremath{\omega}/2)=x. We provide expansions of \ensuremath{\omega}(x) in powers of x and ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ for the regimes of small and large x, respectively, and establish their radii of convergence. The scaling function shows anomalous behavior at small x, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}x, instead of the usual ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}
xz, as x\ensuremath{\rightarrow}0. This is because, even for finite L, the lifetime of the slowest dynamical mode diverges for T\ensuremath{\rightarrow} 0 K. For periodic BC's, with the exception of one system, ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$ is independent of L, and hence ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$=1. The exceptional system, that with an odd number of spins and antiferromagnetic couplings, exhibits frustration at T=0 K, and the scaling function is given by ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\pi}/x
)2${]}^{\mathrm{\ensuremath{-}}1}$. \textcopyright{} 1996 The American Physical Society.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.
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.