Mixed spectra and partially extended states in a two-dimensional quasiperiodic model

We introduce a two-dimensional generalization of the quasiperiodic Aubry-Andre model. Even though this model exhibits the same duality relation as the one-dimensional version, its localization properties are found to be substantially more complex. In particular, partially extended single-particle states appear for arbitrarily strong quasiperiodic modulation. They are concentrated on a network of low-disorder lattice lines, while the rest of the lattice hosts localized states. This spatial separation protects the localized states from delocalization, so no mobility edge emerges in the spectrum. Instead, localized and partially extended states are interspersed, giving rise to an unusual type of mixed spectrum and enabling complex dynamics even in the absence of interactions. A striking example is ballistic transport across the low-disorder lines while the rest of the system remains localized. This behavior is robust against disorder and other weak perturbations. Our model is thus directly amenable to experimental studies and promises fascinating many-body localization properties.