We construct a tight-binding model that hosts both a quasi-periodic nature and marcoscopically-dengenerate zero-energy modes. The model can be regarded as a counterpart of the Aubry-Andr\'{ e}-Harper (AAH) model, which is a paradigmatic example of the quasi-periodic tight-binding model. Our main focus is on the many-body state where the flat-band-like degenerate zero-energy modes are fully occupied. We find a characteristic sublattice dependence of the particle density distribution. Further, by analyzing the hyperuniformity of the particle density distribution, we find that it belongs to the class-I hyperuniform distribution, regardless of the model parameter. We also show that, upon changing the parameter, the finite-energy modes exhibit the same extended-to-localized transition as that for the original AAH model.
Categories: cond-mat.mes-hall cond-mat.dis-nn cond-mat.stat-mech cond-mat.str-el