Title: Analytical derivation and expansion of the anti-Kibble-Zurek scaling in the transverse field Ising model

Abstract: A defect density which quantifies the deviation from the spin ground state characterizes non-equilibrium dynamics during phase transitions. The widely recognized Kibble-Zurek scaling predicts how the defect density evolves during phase transitions. However, it can be perturbed by noise, leading to anti-Kibble-Zurek scaling. In this research, we analytically investigate the effect of Gaussian white noise on the transition probabilities of the Landau-Zener model. We apply this model to the one-dimensional transverse field Ising model and derive an analytical approximation for the defect density. Our analysis reveals that under small noise conditions, the model follows an anti-Kibble-Zurek scaling. As the noise increases, a new scaling behavior emerges, showing higher accuracy than previously reported. Furthermore, we identify the parameters that optimize the defect density based on the new scaling. This allows for the refinement of optimized parameters with greater precision and provides further validations of previously established scaling.