Kardar-Parisi-Zhang and glassy properties in 2D Anderson localization: eigenstates and wave packets

Despite decades of research, the universal nature of fluctuations in disordered quantum systems remains poorly understood. Here, we present extensive numerical evidence that fluctuations in two-dimensional (2D) Anderson localization belongs to the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class. In turn, by adopting the KPZ framework, we gain fresh insight into the structure and phenomenology of Anderson localization itself. We analyze both localized eigenstates and time-evolved wave packets, demonstrating that the fluctuation of their logarithmic density follows the KPZ scaling. Moreover, we reveal that the internal structure of these eigenstates exhibits glassy features characteristic of the directed polymer problem, including the emergence of dominant paths together with pinning and avalanche behavior. Localization is not isotropic but organized along preferential branches of weaker confinement, corresponding to these dominant paths. For localized wave packets, we further demonstrate that their spatial profiles obey a stretched-exponential form consistent with the KPZ scaling, while remaining fully compatible with the single-parameter scaling (SPS) hypothesis, a cornerstone of Anderson localization theory. Altogether, our results establish a unified KPZ framework for describing fluctuations and microscopic organization in 2D Anderson localization, revealing the glassy nature of localized states and providing new understanding into the universal structure of disordered quantum systems.