Pythagoras superposition principle for localized eigenstates of two-dimensional moire lattices

Moire lattices are aperiodic systems formed by a superposition of two periodic lattices with a relative rotational angle. In optics, the photonic moire lattice has many appealing properties such as its ability to localize light, thus attracting much attention on exploring features of such a structure. One fundamental research area for photonic moire lattices is the properties of eigenstates, particularly the existence of localized eigenstates and the localization-to-delocalization transition in the energy band structure. Here we propose an accurate algorithm for the eigenproblems of aperiodic systems by combining plane-wave discretization and spectral indicator validation under the higher-dimensional projection, allowing us to explore energy bands of fully aperiodic systems. A localization-delocalization transition regarding the intensity of the aperiodic potential is observed and a Pythagoras superposition principle for localized eigenstates of two-dimensional moire lattices is revealed by analyzing the relationship between the aperiodic system and its corresponding periodic eigenstates. This principle sheds light on exploring the physics of localizations for moire lattices.

Automated summary

A research team proposes an accurate algorithm for the eigenproblems of aperiodic systems, allowing exploration of energy bands in fully aperiodic systems. The relationship between the intensity of the aperiodic potential and the localization-to-delocalization transition is observed, and a Pythagoras superposition principle for localized eigenstates of two-dimensional moire lattices is revealed. This principle sheds light on exploring the physics of localizations for moire lattices.