Localization and dynamics in one-dimensional and two-dimensional Gaussian disordered quantum spin systems

We propose a Gaussian disorder to study many-body localization (MBL) in one-dimensional quantum system and coupled system-environment models. Unlike uniform disorder, Gaussian disorder is controlled by the standard deviation $\sigma$ (driving MBL transitions) and the mean $\mu$ (causing global energy shifts). By calculating the von Neumann entropy, mean gap ratio, and dynamical indicators, we elucidate the ergodic-to-MBL transition as $\sigma$, with Gaussian disorder exhibiting more pronounced finite-size critical point drift effects compared to conventional random disorder. Through finite-size scaling analysis, it shows that there are no significant differences between the two disorder types at our assumed system sizes. We construct two system-environment coupling configurations (ladder and staggered), showing that under weak coupling, the system and environment evolve independently, whereas strong coupling induces cooperative localization. The ladder configuration shows a strong dependence on the initial state, which may lead to either the loss or retention of information, while no such dependence is observed in the staggered configuration. This can be explained by the changes in energy density of the initial state caused by variations in different model structures. When treating the system-environment as a unified two-dimensional ladder, asymmetric disorder on the two chains can drive a global MBL phase transition, with slightly different behaviors compared to symmetric disorder. Additionally, we emphasize the necessity of correctly selecting the initial state when characterizing the phase transition through dynamic indicators.