Despite their potential similarity between the mosaic Aubry-André (MAA) and AA models, the MAA model allows mobility edges (MEs), whereas the AA model does not. Here we develop a new double quasiperiodic MAA (DMAA) model consisting of one primitive MAA with nonzero even-site potentials and the other modified one with both nonzero odd-site potentials and a tunable amplitude factor, to reveal how localization transitions evolve from MAA to AA models. Interplays and competitions among the extended, critical and localized states arising from superpositions of double quasi-periodic MAA potentials enable new twice and multiple localization-delocalization transitions besides the original single localization transition. Our numerical calculations on inverse participation ratio, normalized participation ratio, fractal dimension and real-space wavefunction distribution confirm such localization features. The continuum model simulations on the experimental polariton modes also yield consistent results and hence validate their experimental feasibility. The constructed DMAA model provides a new framework for studying the localization transition processes between two analogous quasiperiodic models and broadens the understanding of Anderson localization.
We study the single-particle properties of two-dimensional quasicrystals where the underlying geometry of the tight-binding lattice is crystalline but the on-site potential is quasicrystalline. We will focus on the two-dimensional (2D) generalized Aubry-André model which has a varying form to its quasiperiodic potential, through a deformation parameter and varied irrational periods of cosine terms, which allows a continuous family of on-site quasicrystalline models to be studied. We show that the 2D generalized Aubry-André model exhibits many single-particle mobility edges which we confirm for finite systems and supports critical states across large parameter regions. Critical states are neither fully localized nor extended. We observe that diagonal quasicrystalline models can support many energy intervals of critical states in the spectrum while stabilizing both localized and extended states in other energy intervals; we refer to these as anomalous mobility edges. We show that critical states are present independent of system size through a scaling analysis of the inverse participation ratio and that they are present in spectra that also contain extended and localized states, confirming that at least one anomalous mobility edge is present. Due to this, these models exhibit anomalous diffusion of initially localized states across the majority of parameter regions, including deep in the normally localized regime. The presence of critical states in large parameter regimes and throughout the spectrum will have consequences for the many-body properties of quasicrystals, including the formation of the Bose glass and the potential to host a many-body localized phase. Published by the American Physical Society 2024
In this paper, we explore quantum criticality in the disordered Aubry-Andr\'e (AA) model. For the pure AA model, it is well known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $\mathrm{\ensuremath{\Delta}}$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $\ensuremath{\xi}$ scales with $\mathrm{\ensuremath{\Delta}}$ as $\ensuremath{\xi}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}{\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}}$ with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}$ a new critical exponent, which is estimated to be ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}\ensuremath{\approx}0.46$. This value is remarkably different from the counterparts for both the pure AA model and the Anderson model. Moreover, rich critical phenomena are discovered in the critical region spanned by the quasiperiodic and the disordered potentials. In particular, in the extended phase side, we show that the scaling theory satisfies a hybrid scaling form as a result of the overlap between the critical regions of the AA model and the Anderson localization.
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.
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.
We propose a one-dimensional generalized Aubry-Andr\'e-Harper (AAH) model with off-diagonal hopping and staggered on-site potential. We find that the localization transitions could be multiple reentrant with the increasing of staggered on-site potential. The multiple localization transitions are verified by the quantum static and dynamic measurements such as the inverse or normalized participation ratios, fractal dimension, and survival probability. Based on the finite-size scaling analysis, we also obtain an interesting intermediate phase where the extended, localized, and critical states are coexistent in certain regimes of model parameters. These results are quite different from those in the generalized AAH model with off-diagonal hopping, and can help us to find novel quantum phases and new localization phenomena in the disordered systems.
One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.