Quantum phase transition between hyperuniform density distributions

We study an electron distribution under a quasiperiodic potential in light of hyperuniformity, aiming to establish classification and analysis method for aperiodic but orderly density distributions realized in, e.g., quasicrystals. Using the Aubry-Andr\'e-Harper model, we first reveal that electron-charge changes its character as increased alters eigenstates from extended localized ones. While these charge are characterized by neither multifractality nor translational-symmetry breaking, they hyperuniformity class order metric. find nontrivial relationship between states at Fermi level, charge-distribution histogram, class. The change different occurs first-order phase transition except electron-hole symmetric point, where is third order. Moreover, generalize metric function, capture more detailed features distribution, some analogy with generalization fractal dimension multifractal one.