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arXiv:2404.14971 · Physical review. B./Physical review. B

Stark localization near Aubry-André criticality

Ayan Sahoo, Aitijhya Saha, Debraj Rakshit

Published Jan 9, 2025• 11 citations
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Abstract

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, ${F}_{Q}$, 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, ${F}_{Q}$ 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.

DOI
10.1103/physrevb.111.024205

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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.

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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 hhh, the localization length \ensuremathζ\ensuremath{\zeta}\ensuremathζ scales as \ensuremathζ\ensuremath∝h\ensuremath−\ensuremathν\ensuremath{\zeta}\ensuremath{\propto}{h}^{\ensuremath{-}\ensuremath{\nu}}\ensuremathζ\ensuremath∝h\ensuremath−\ensuremathν with \ensuremathν\ensuremath≈0.29\ensuremath{\nu}\ensuremath{\approx}0.29\ensuremathν\ensuremath≈0.29, which is different from both the pure AA model (\ensuremathν=1)(\ensuremath{\nu}=1)(\ensuremathν=1) and Stark model (\ensuremathν\ensuremath≈0.33)(\ensuremath{\nu}\ensuremath{\approx}0.33)(\ensuremathν\ensuremath≈0.33). The IPR in this case scales as IPR \ensuremath∝hs\ensuremath{\propto}{h}^{s}\ensuremath∝hs with s\ensuremath≈0.096s\ensuremath{\approx}0.096s\ensuremath≈0.096, which is again significantly different from both the pure AA model (s\ensuremath≈0.33)(s\ensuremath{\approx}0.33)(s\ensuremath≈0.33) and Stark model (s\ensuremath≈0.33)(s\ensuremath{\approx}0.33)(s\ensuremath≈0.33). The energy gap, \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ, scales as E\ensuremath∝h\ensuremathνzE\ensuremath{\propto}{h}^{\ensuremath{\nu}z}E\ensuremath∝h\ensuremathνz, where z\ensuremath≈2.37z\ensuremath{\approx}2.37z\ensuremath≈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{F}_{Q}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{F}_{Q}FQ​ scales with the system size as FQ\ensuremath∝L\ensuremathβ{F}_{Q}\ensuremath{\propto}{L}^{\ensuremath{\beta}}FQ​\ensuremath∝L\ensuremathβ, where \ensuremathβ\ensuremath≈6.7\ensuremath{\beta}\ensuremath{\approx}6.7\ensuremathβ\ensuremath≈6.7, which is greater in comparison to the pure Stark case, for which \ensuremathβ\ensuremath{\beta}\ensuremathβ 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 explore quantum criticality and Kibble-Zurek scaling (KZS) in the Aubry-Andr\'e-Stark (AAS) model, where the Stark field of strength $\ensuremath{\varepsilon}$ is added onto the one-dimensional quasiperiodic lattice. We perform scaling analysis and numerical calculations of the localization length, the inverse participation ratio (IPR), and the energy gap between the ground and first excited states to characterize critical properties of the delocalization-localization transition. Remarkably, our scaling analysis shows that, near the critical point, the localization length $\ensuremath{\xi}$ scales with $\ensuremath{\varepsilon}$ as $\ensuremath{\xi}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.3$, a new critical exponent for the AAS model, which is different from the counterparts for both the pure Aubry-Andr\'e (AA) model and the pure Stark model. The IPR I\mathcal{I}I scales as $\mathcal{I}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{s}$ with the critical exponent $s\ensuremath{\approx}0.098$, which is also different from both of the pure models. The energy gap $\mathrm{\ensuremath{\Delta}}E$ scales as $\mathrm{\ensuremath{\Delta}}E\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{\nu}z}$ with the same critical exponent $z\ensuremath{\approx}2.374$ as that for the pure AA model. We further reveal hybrid scaling functions in the overlap between the critical regions of the Anderson and Stark localizations. Moreover, we investigate the driven dynamics of the localization transitions in the AAS model. By linearly changing the Stark (quasiperiodic) potential, we calculate the evolution of the localization length and the IPR, and we study their dependence on the driving rate. We find that the driven dynamics from the ground state is well described by the KZS with the critical exponents obtained from the static scaling analysis. When both the Stark and the quasiperiodic potentials are relevant, the KZS form includes the two scaling variables. This work extends our understanding of critical phenomena on localization transitions and generalizes the application of the KZS to hybrid models.
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 χχχ and the eigenstate fractal dimension d(1)d^{(1)}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 zzz, 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=3z=3z=3.
The localization transition can be exploited as a resource for achieving quantum-enhanced sensitivity in parameter estimation. We demonstrate that by employing different classes of localization inducing potentials, one can significantly enhance the precision of parameter estimation. Specifically, we focus on the precision measurement of the Stark strength parameter encoded in the low- and high-energy eigenstates of a one-dimensional fermionic lattice under the influence of Aubry-André-Harper localization-delocalization transition. For the ground state, we consider the single-particle system, in addition to the system at half filling. Our work reveals that Quantum Fisher Information (QFI) offers superior scaling with respect to the system size compared to the pure Stark case, leading to a better parameter estimation. However, experimental measurement of the QFI based on fidelity in a multibody system is a significant challenge. To address this, we suggest experimentally relevant operators that can be utilized to achieve precision surpassing the Heisenberg Limit (HL) or can even saturate the QFI scaling. These operators, relevant for practical experimental setups, provide a feasible pathway to harness the advantages offered by the localization-delocalization transition by exploiting two distinct localizing potentials for quantum-enhanced parameter estimation.
We have investigated scaling properties near the quantum critical point between the extended phase and the critical phase in the Aubry-Andr\'e-Harper model with ppp-wave pairing, which have rarely been exploited as most investigations focus on the localization transition from the critical phase to the localized phase. We find that the spectrum-averaged entanglement entropy and the generalized fidelity susceptibility act as eminent universal order parameters of the corresponding critical point without gap closing. We introduce a Widom scaling Ansatz for these criticality probes to develop a unified theory of critical exponents and scaling functions. We thus extract the correlation-length critical exponent $\ensuremath{\nu}$ and the dynamical exponent zzz through the finite-size scaling given the system size increase in the Fibonacci sequence. The retrieved values of $\ensuremath{\nu}\ensuremath{\simeq}1.000$ and $z\ensuremath{\simeq}3.610$ indicate that the transition from the extended phase to the critical phase belongs to a different universality class from the localization transition. Our approach sets the stage for exploring the unconventional quantum criticality and the associated universal information of quasiperiodic systems in state-of-the-art quantum simulation experiments.
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 zzz 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 LLL. Even though for infinite LLL 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 zzz 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.