Quantum Hyperuniformity and Quantum Weight

Extending hyperuniformity from classical to quantum fluctuations in electron systems yields a framework that identifies quantum phase transitions and reveals underlying gap structures through the quantum weight. We study long-wavelength fluctuations of many-body ground states through the charge-density structure factor by incorporating intrinsic quantum fluctuations into hyperuniformity. Although charge fluctuations at zero temperature are generally suppressed by particle-number conservation, their long-wavelength scaling reveals distinct universal behaviors that define quantum hyperuniformity classes. By exemplifying the Aubry-Andre model, we find that gapped, gapless, and localized-critical-extended phases are sharply distinguished by the quantum hyperuniformity classes. Notably, at the critical point, multifractal wave functions generate anomalous scaling behavior. We further show that, in quantum-hyperuniform gapped phases, the quantum weight provides a quantitative measure of the gap size through a universal power-law scaling. Along with classical hyperuniformity, quantum hyperuniformity serves a direct fingerprint of quantum criticality and a practical probe of quantum phase transitions in aperiodic electron systems.