Computational study of the spectral behaviour of different isospectrally patterned lattices

We perform a computational spectral analysis of isospectrally patterned lattices (IPL) that consist of coupled degenerate cells, being uniquely parametrized by a set of phases. We focus on two-dimensional cells and explore symmetric as well as asymmetric IPL. A tunable fraction of localized vs. delocalized eigenstates belonging to the three subdomains of the corresponding energy bands is demonstrated and analyzed with different measures of localization. For the asymmetric case the center of localization can be shifted arbitrarily by shifting the underlying phase grid. Introducing a complete phase revolution leads for low and high energies to two well-separated branches of localized states which finally merge with increasing energy into the branch of delocalized states. Remarkably, the localized states appear in near-degenerate pairs and this near-degeneracy is lifted upon entering the delocalization regime. A corresponding generalization to several phase revolutions is provided showing a characteristic nodal pattern among the near-degenerate eigenstates.