Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems

A moiré potential—the superposition of two periodic potentials with different wavelengths—will either introduce a new periodicity into a system if the two potentials are commensurate or force the system to be quasiperiodic if they are not. Here we demonstrate that quasiperiodicity can change the ground-state properties of one-dimensional moiré systems with respect to their periodic counterparts. We show that although narrow bands play a role in enhancing interactions, for both commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the state enabled by quasiperiodicity has contributions from electronic states with a very large number of wavevectors. This quasi-fractal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and highlight that multifractal non-interacting phases might be particularly promising parent states.