As opposed to random disorder, which localizes single-particle wave functions in one dimension (1D) at arbitrarily small disorder strengths, there is a localization-delocalization transition for quasiperiodic disorder in the 1D Aubry-Andr\'e model at a finite disorder strength. On the single-particle level, many properties of the ground state at criticality have been revealed by applying a real-space renormalization-group scheme; the critical properties are determined solely by the continued-fraction expansion of the incommensurate frequency of the disorder. Here, we investigate the many-particle localization-delocalization transition in the Aubry-Andr\'e model with and without interactions. In contrast to the single-particle case, we find that the critical exponents depend on a Diophantine equation relating the incommensurate frequency of the disorder and the filling fraction which generalizes the dependence, in the single-particle spectrum, on the continued-fraction expansion of the incommensurate frequency. This equation can be viewed as a generalization of the resonance condition in the commensurate case. When interactions are included, numerical evidence suggests that interactions may be irrelevant at at least some of these critical points, meaning that the critical exponent relations obtained from the Diophantine equation may actually survive in the interacting case.
In this work, we investigate the Stark localization near the Aubry-Andr\'e (AA) critical point. We perform careful studies for reporting system-dependent parameters, such as localization length, inverse participation ratio (IPR), and energy gap between the ground and first excited state, for characterizing the localization-delocalization transition. We show that the scaling exponents possessed by these key descriptors of localization are quite different from that of a pure AA model or Stark model. Near the critical point of the AA model, in the presence of Stark field of strength
h, the localization length $\ensuremath{\zeta}$ scales as $\ensuremath{\zeta}\ensuremath{\propto}{h}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.29$, which is different from both the pure AA model $(\ensuremath{\nu}=1)$ and Stark model $(\ensuremath{\nu}\ensuremath{\approx}0.33)$. The IPR in this case scales as IPR $\ensuremath{\propto}{h}^{s}$ with $s\ensuremath{\approx}0.096$, which is again significantly different from both the pure AA model $(s\ensuremath{\approx}0.33)$ and Stark model $(s\ensuremath{\approx}0.33)$. The energy gap, $\mathrm{\ensuremath{\Delta}}$, scales as $E\ensuremath{\propto}{h}^{\ensuremath{\nu}z}$, where $z\ensuremath{\approx}2.37$, which is however the same as the pure AA model. Finally, we discuss how invoking a criticality-inducing additional control parameter may help in designing better many-body quantum sensors. Quantum critical sensors exploit the venerability of the wave function near the quantum critical point against small parameter shifts. By incorporating a control parameter in the form of the quasiperiodic field, i.e., the AA potential, we show a significant advantage can be drawn in estimating an unknown parameter, which is considered here to be the Stark weak field strength, with high precision. We compute and extract the scaling exponent associated with the quantum Fisher information,
FQ, which provides a bound on the uncertainty associated with measuring an unknown parameter. We demonstrate that by a control field in the form of the quasiperiodic potential, and setting it at AA criticality,
FQ scales with the system size as ${F}_{Q}\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, where $\ensuremath{\beta}\ensuremath{\approx}6.7$, which is greater in comparison to the pure Stark case, for which $\ensuremath{\beta}$ is known to be 5.8. We, thus, pitch a generic idea of invoking quantum-criticality-inducing additional control parameters for obtaining enhanced quantum advantage in many-body critical sensors.
The Aubry-Andr\'e one-dimensional lattice model describes a particle hopping in a pseudorandom potential. Depending on its strength $\ensuremath{\lambda}$, all eigenstates are either localized $(\ensuremath{\lambda}>1)$ or delocalized $(\ensuremath{\lambda}<1)$. Near the transition, the localization length diverges like $\ensuremath{\xi}\ensuremath{\sim}{(\ensuremath{\lambda}\ensuremath{-}1)}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=1$. We show that when the particle is initially prepared in a localized ground state and the potential strength is slowly ramped down across the transition, then---in analogy with the Kibble-Zurek mechanism---it enters the delocalized phase having finite localization length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+z\ensuremath{\nu})}$. Here ${\ensuremath{\tau}}_{Q}$ is the ramp/quench time and
z is a dynamical exponent. At $\ensuremath{\lambda}=1$ we determine $z\ensuremath{\simeq}2.37$ from the power-law scaling of an energy gap with a lattice size
L. Even though for infinite
L the model is gapless, we show that the gap that is relevant for excitation during the ramp remains finite. Close to the critical point it scales like ${\ensuremath{\xi}}^{\ensuremath{-}z}$ with the value of
z determined by the finite-size scaling. It is the gap between the ground state and the lowest of those excited states that overlaps with the ground state enough to be accessible for excitation. We propose an experiment with a noninteracting BEC to test our prediction. Our hypothesis is further supported by considering a generalized version of the Aubry-Andr\'e model possessing an energy-dependent mobility edge.
We numerically study spectral and wave-function properties, as well as time evolution, in finite-size commensurate approximations of the 3D Aubry-André model. We explicitly show, through scaling analysis, the existence of two critical points. One of the critical points is a localization-delocalization transition in real-space, similar to the critical point in the 3D Anderson model. The other is a localization-delocalization transition in a dual momentum space, which does not have any analog in the 3D Anderson model. We estimate their values from spectral measures, such as the gap ratio and the spectral form factor, as well as from wave-function measures, such as the inverse participation ratio and the participation entropy. Additionally, we demonstrate that both critical points exhibit hallmarks of critical points between quantum ergodicity and localization. First, the eigenstates and the eigenspectrum at the critical point fulfill the mutual universal relation between the eigenspectrum compressibility
χ and the eigenstate fractal dimension
d(1) [S. Jiricek et al., Proc. Nat. Acad. Sci. 123, e2518027123 (2026)]. Second, the dynamics of the wave packet at the critical point is governed by the critical dynamical exponent
z, characterized by the spectral fractal dimension and the dimensionality of the lattice [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 135, 060401 (2025)]. Thus, similarly to the 3D Anderson model, the critical dynamical exponent is sub-diffusive with the value of
z=3.