Isospectrally patterned lattices

We introduce and explore patterned lattices consisting of coupled isospectral cells that vary across the lattice. The isospectrality of the cells is encapsulated in the phase that characterizes each cell and can be designed at will such that the lattice exhibits a certain phase gradient. Focusing on the specific example of a constant phase gradient on a given finite phase interval we show that the resulting band structure consists of three distinct energy domains with two crossover edges marking the transition from single center localized to delocalized states and vice versa. The characteristic localization length emerges due to a competition of the involved phase gradient on basis of a local rotation and the coupling between the cells which allows us to illuminate the underlying localization mechanism and its evolution. The fraction of localized versus delocalized eigenstates can be tuned by changing the phase gradient between the cells of the lattice. We outline the perspectives of investigation of this novel class of isospectrally patterned lattices.