Short-time dynamics and critical behavior of the three-dimensional site-diluted Ising model

Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations $p=0.95$ and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.

Figure 1

Time evolution of the magnetization $m\left(t\right)$ from the initial state with magnetization $m_0=0.03$ at $T_c=3.49948$ as a result of Monte Carlo simulation of samples with spin concentration $p=0.8$ and with linear size $L=128$.

Figure 2

Time evolution of the magnetization $m\left(t\right)$ is plotted on a log-log scale for samples with spin concentrations (a) $p=0.95$ and (b) $p=0.8$ at $T=T_c\left(p\right)$ (curves 1) and at $T=T_c\left(p\right)\mp \Delta T$, with $\Delta T=0.005$ (curves 2 and 3). In insets, curves 4 correspond to pure Ising model.

Figure 3

Time evolution of the logarithmic derivative of the magnetization ${\partial }_{\tau }\text{ln}m\left(t,\tau \right){\mid }_{\tau =0}$ with respect to $\tau$ is plotted on a log-log scale for samples with spin concentrations (a) $p=0.95$ and (b) $p=0.8$.

Figure 4

Time evolution of the cumulant $U_2\left(t\right)$ is plotted on a log-log scale at $T=T_c\left(p\right)$ for samples with spin concentrations (a) $p=0.95$ and (b) $p=0.8$.

Figure 5

Dependence of the mean-square errors $\sigma$ of the (a) fits for the magnetization, (b) logarithmic derivative of the magnetization, and (c) cumulant as a function of the exponents $\beta /\nu z$, $1/\nu z$, and $d/z$ for $p=0.8$.

Figure 6

Time evolution of the magnetization $m\left(t\right)$ for different values of the initial magnetization $m_0=0.01$ (1); 0.02 (2); 0.03 (3), plotted on a log-log scale for samples with spin concentration $p=0.8$.

Figure 7

(a) Time evolution of the second moment $m^{\left(2\right)}$ and (b) the correlation function $A\left(t\right)$ for $L=128$ with the initial magnetization $m_0=0.0001$.