Quantum phase transition between hyperuniform density distributions

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