Quantum Hyperuniformity and Quantum Weight

TL;DR

The work extends hyperuniformity to quantum fluctuations by defining a quantum structure factor $S_Q(q)$ that isolates quantum contributions to density fluctuations and classifies ground states into quantum hyperuniformity (QHU) classes I, II, and III. Using the Aubry–André model and mobility-edge variants, it shows that CHU and QHU provide complementary fingerprints of spectral gaps, localization, and multifractal criticality, with the quantum weight $K$ linking quadratic infrared scaling to gap size. At criticality, multifractal wavefunctions yield anomalous $S_Q(q)$ with nonuniversal exponents, while in gapped and localized regimes $S_Q(q)$ follows quadratic scaling, allowing gap estimation from $K$. The results offer a practical, experimentally accessible framework to diagnose quantum phase transitions in aperiodic systems via ground-state density fluctuations and quantum geometry.

Abstract

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.

Figures (7)

• Figure 1: (a) Spectrum and (b) phase diagram of CHU class for the Aubry–André model. Note that tiny gaps are present but invisible in this resolution. See Supplemental Material supplementQWAA for a more detailed structure. (c) Scaling behaviors of $S_Q$ for various $\lambda$ values. Solid lines (left axis) indicate the results for a Fermi level crossing a band, of which the scaling exponents are $1$, $0.618$, and $2$ for $\lambda=1,2$, and $3$, respectively. Dashed lines (right axis) indicate the results for a Fermi level inside a gap, of which the scaling exponent is $2$ for all $\lambda$. (d) Scaling exponent, $\nu$, plotted against $\lambda$. The black open square indicates the results for the Fermi level inside a gap, while the other symbols represent those for distinct Fermi levels crossing a band. The inset is drawn to emphasize a diversification of $\nu$ at $\lambda=2$. (e) The landscape of $\nu$ plotted against $\lambda$ and the Fermi energy $E_F$. (f) Phase diagram of total hyperuniformity classes incorporating classical and quantum fluctuations. The red, black and skyblue colors represent class I, II and III, respectively. The system size is $N=17711$.
• Figure 2: (a) The landscape of quantum weight $K$ plotted against $\lambda$ and $E_F$. Black indicates infinite $K$. (b) Log--log plot of $K^{-1}$ for the filling fractions, $\varphi^{s}=F_s\varphi+F_{s-1}$ with integers $s\ge 1$, which correspond to the gaps of the Aubry–André model, as a function of $\lambda$. These quantum weights exhibit scaling behavior, $K\propto\vert \lambda\vert^{-\mu}$. (c) The gap size $\Delta E$ at $\lambda=1$ for different gaps as a function of the scaling exponent $\mu$, which show a relation $\Delta E_{\lambda=1} \propto e^{-\alpha\mu^{2/3}}$. (d) Gap sizes at different filling fractions as functions of $K^{-1}$, normalized by their values at $K=1$, which show $K^{-\beta}$ universal scaling. Each color represents a gap for $1.5\le\lambda\le2$ at a filling fraction $a\varphi + b$ with a different integer $a$. The system size is $N=17711$.
• Figure S3: Log--log plot of the averaged long-range correlation function, $C(r)$ for different $\nu$ values. Here, the Fermi levels are not in the gap. Smaller $\nu$ corresponds to a slower decay of $C(r)$. The system size is $N=17711$. $t=1$.
• Figure S4: Gap sizes at different filling fractions as functions of $K$. Here, $\lambda> 2$. Each color represents a gap at a filling fraction $a\varphi + b$ with a different integer $a$. The system size is $N=17711$. $t=1$. Here, we use $\varphi=10946/17711$.
• Figure S5: (a) Landscape of the finite-size fractal dimension of the eigenstates of the Hamiltonian in Eq.\ref{['AASSH']}. (b) Scaling exponent of $S_Q$ for small-$q$, $\nu$ as the function of Fermi energy and $\lambda$. The system size is $N=10946$. $t_1=1$ and $t_2=0.3$.