  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.

Strongly Correlated Electrons;Disordered Systems and Neural Networks;Quantum Gases;Statistical Mechanics