Short-Time Dynamics for the Three-Dimensional O(4) Model

We perform a numerical study of the short-time dynamics for the O(4) spin model in three dimensions. Using a heat-bath time evolution algorithm we obtain the first and the second magnetization moments. With this quantities we estimate the dynamic critical exponents $\theta,\, \theta_g$ and $y$. The $\theta$ exponent is related to the magnetization short-time anomalous behavior. Indeed, from dynamic scaling relation it is possible to observe universality and scaling behavior in the beginning of system evolution after quenching the system from high temperature to the Curie temperature. To measure the dynamic critical exponent $\theta$ we perform the simulations at $T_c = 1.0683$ for different (sharp) values of the $m_0$ initial magnetization. We fit a power law in the magnetization data. In order to avoid the extrapolation and the cumulative error in $\theta$ we use the time correlation function. Then we have the $\theta$ measure from two approaches. The results agree with literature's analytical calculations. The dynamic critical exponent for the magnetization second moment ($y$) is calculated from $m_0=0$ (not sharp). The dynamic critical exponent $\theta_g$ related with the magnetization probability does not change the signal of its initial value at time t. For a direct measurement of the $z$ critical exponent we use the mix method.