Coexistence of Localized and Extended States in Disordered Systems

It is well known that in disordered systems, Anderson localized states and extended states do not coexist at the same energy. Here we propose a simple mechanism to achieve the coexistence of localized and extended states both spectrally and spatially in a class of disordered quasi-1D and quasi-2D systems. The systems are partially disordered in a way that a few bands of extended states always exists, not affected by the randomness, whereas the states in all other bands become localized. The bands of extended states can overlap with those bands of localized states both in energy and in space, achieving the aforementioned coexistence. We demonstrate such coexistence in disordered multi-chain and multi-layer systems. Such systems can be experimentally realized on multiple platforms such as cold atoms and optics.