Fraction of delocalized eigenstates in the long-range Aubry-André-Harper model

Abstract

We uncover a systematic structure in the single-particle phase diagram of the quasiperiodic Aubry-Andr\\ e-Harper (AAH) model with power-law hoppings ($\\ensuremath{\\sim}\\frac{1}{{r}^{\\ensuremath{\\sigma}}}$) when the quasiperiodicity parameter is chosen to be a member of the metallic mean family of irrational Diophantine numbers. In addition to the fully delocalized and localized phases, we find a coexistence of multifractal (localized) states with the delocalized states for $\\ensuremath{\\sigma}l1$ ($\\ensuremath{\\sigma}g1$). The fraction of delocalized eigenstates in these phases can be obtained from a general sequence, which is a manifestation of a mathematical property of the metallic mean family. The entanglement entropy of the noninteracting many-body ground states respects the area law if the Fermi level belongs in the localized regime while logarithmically violating it if the Fermi level belongs in the delocalized or multifractal regimes. The prefactor of the logarithmically violating term is significantly larger in the delocalized phase in comparison to that in the multifractal phase. Entanglement entropy shows the area law even in the delocalized regime for special filling fractions, which are related to the metallic means.