  In this paper, we continue to study the Calabi flow on complex tori. We
develop a new method to obtain an explicit bound of the curvature of the Calabi
flow. As an application, we show that when $n=2$, the Calabi flow starting from
a weak K\"ahler metric will become smooth immediately. It implies that in our
settings, the weak minimizer of the Mabuchi energy is a smooth one.
