With Monte Carlo methods, we simulate dynamic relaxation processes of the two-dimensional $XY$ model at the Berezinskii-Kosterlitz-Thouless phase transition temperature and below, starting from both ordered and disordered initial states. The two-time correlation function $A(t^{\prime },t)$ is measured, and aging phenomena are investigated. The power-law correction in the spatial correlation length $\xi \left(t\right)$ for relaxation with an ordered initial state and the logarithmic correction for relaxation with a disordered initial state are carefully analyzed. The scaling functions of $A(t^{\prime },t)$ are then extracted.

• Received 6 August 2006

DOI:https://doi.org/10.1103/PhysRevE.75.040104