With dynamic Monte Carlo simulations,aging phenomena of two-dimensional Ising model and dynamic XY model were studied numerically.Systems were quenched to their critical point from a disordered initial state.The autocorrelation function was calculated by heat-bath algorithm.Critical exponents were also calculated.It was confirmed that for the 2D XY model,there is a logarithmic correction to scaling.The universality proposal and their scale relation in aging phenomena were predicted.The scale relation agreed with renormalization group calculations.
Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L\ifmmode\times\else\texttimes\fi{}L lattices, the scaling behavior of the susceptibility \ensuremath{\chi} and correlation length \ensuremath{\xi} at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the finite-size scaling region and (ln \ensuremath{\xi}${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to
5122 we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.