Fraction of delocalized eigenstates in the long-range Aubry-Andr\'e-Harper model

We uncover a systematic structure in the single particle phase-diagram of the quasiperiodic Aubry-Andr\'e-Harper(AAH) model with power-law hoppings ($\sim \frac{1}{r^\sigma}$) 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 co-existence of multifractal (localized) states with the delocalized states for $\sigma<1$ ($\sigma>1$). 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 logarithmically violating term shows interesting behavior in different phases. Entanglement entropy shows the area-law even in the delocalized regime for special filling fractions, which are related to the metallic means.