Driven dynamics of localization phase transition in the Aubry-Andr\'{e} model with a initial gapless state

Recently, the driven dynamics of localization phase transitions have garnered growing interest.However, studies so far have mainly considered initial localized states, whose driven dynamics follow Kibble-Zurek mechanism (KZM).In this study, we investigate the non-equilibrium dynamics of the localization phase transition in the Aubry-Andr\'e (AA) model starting from a gapless extended state, which violates the adiabatic-impulse scenario of KZM.By linearly driving the quasiperiodic potential strength across the critical point, we numerically simulate the driven dynamics and analyze the scaling behavior of both the inverse participation ratio ($\mathcal{I}$) and the dynamic deviation from the instantaneous ground state energy $(\mathcal{D})$.We demonstrate that the driven dynamics with initial extended states satisfies the finite-time scaling (FTS) criterion.The scaling functions governing the driven dynamics of both $\mathcal{I}$ and $\mathcal{D}$ have been derived and numerically validated.We found that the scaling functions of the two order parameters exhibit significant differences at large $R$ and small $R$, and also differ considerably from the scaling functions when the initial state is localized, highlighting the crucial role of initial state behavior.The established scaling laws remain robust across a wide range of system sizes and driving rates, providing testable predictions for experimental realizations.