Out-of-equilibrium finite-size scaling in generalized Kibble-Zurek protocols crossing quantum phase transitions in the presence of symmetry-breaking perturbations

We study the effects of symmetry-breaking perturbations in out-of-equilibrium quantum dynamics many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across critical regime associated with continuous transition (CQT). For this purpose, we analyze arising from generalized Kibble-Zurek (KZ) protocols, within dynamic renormalization-group framework allowing for finite-size systems. show that time dependence generic observables develops an scaling (FSS) behavior, interplay between timescale ${t}_{s}$ parameter variations KZ protocol, size $L$ system, and strength $h$ perturbation, limit large $L$. Moreover, arguments based on first-order adiabatic approximation slow allow us to characterize approach regimes some limits model parameters (for example, when take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions nonadiabatic behaviors limits. This FSS is supported numerical analyses paradigmatic Ising chain along transverse field crossing its CQT, presence static longitudinal breaking ${\mathbb{Z}}_{2}$ symmetry.