Localized and extended phases in square moir\'e patterns

Random defects do not constitute the unique source of electron localization in two dimensions. Lattice quasidisorder generated from two inplane superimposed rotated, main and secondary, square lattices, namely monolayers where moir\'e patterns are formed, leads to a sharp localized to delocalized single-particle transition. This is demostrated here for both, discrete and continuum models of moir\'e patterns that arise as the twisting angle $\theta$ between main and secondary lattices is varied in the interval $[0, \pi/4]$. Localized to delocalized transition is recognized as the moir\'e patterns depart from being perfect square crystals to non-crystalline structures. Extended single-particle states were found for rotation angles associated with Pythagorean triples that produce perfectly periodic structures. Conversely, angles not arising from such Pythagorean triples lead to non-commensurate or quasidisordered structures, thus originating localized states. These conclusions are drawn from a stationary analysis where the standard IPR parameter measuring localization allowed us to detect the transition. While both, ground state and excited states were analyzed for the discrete model, where the secondary lattice was considered as a perturbation of the main one, the sharp transition was tracked back for the fundamental state in the continuous scenario where the secondary lattice is not a perturbation any more.