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arXiv:1811.05496 · Physical review. B./Physical review. B

Kibble-Zurek mechanism with a single particle: Dynamics of the localization-delocalization transition in the Aubry-André model

Aritra Sinha, Marek M. Rams, Jacek Dziarmaga

Published Mar 18, 2019• 58 citations
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Abstract

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.

DOI
10.1103/physrevb.99.094203

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The Aubry-Andr'e one-dimensional lattice model describes a particle hopping in a pseudorandom potential. Depending on its strength \ensuremathλ\ensuremath{\lambda}\ensuremathλ, all eigenstates are either localized (\ensuremathλ>1)(\ensuremath{\lambda}>1)(\ensuremathλ>1) or delocalized (\ensuremathλ<1)(\ensuremath{\lambda}<1)(\ensuremathλ<1). Near the transition, the localization length diverges like \ensuremathξ\ensuremath∼(\ensuremathλ\ensuremath−1)\ensuremath−\ensuremathν\ensuremath{\xi}\ensuremath{\sim}{(\ensuremath{\lambda}\ensuremath{-}1)}^{\ensuremath{-}\ensuremath{\nu}}\ensuremathξ\ensuremath∼(\ensuremathλ\ensuremath−1)\ensuremath−\ensuremathν with \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=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 \ensuremathξ\ifmmode^\elseˆ\fi\ensuremath∼\ensuremathτQ\ensuremathν/(1+z\ensuremathν)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+z\ensuremath{\nu})}\ensuremathξ\ifmmode^\elseˆ\fi​\ensuremath∼\ensuremathτQ\ensuremathν/(1+z\ensuremathν)​. Here \ensuremathτQ{\ensuremath{\tau}}_{Q}\ensuremathτQ​ is the ramp/quench time and zzz is a dynamical exponent. At \ensuremathλ=1\ensuremath{\lambda}=1\ensuremathλ=1 we determine z\ensuremath≃2.37z\ensuremath{\simeq}2.37z\ensuremath≃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 \ensuremathξ\ensuremath−z{\ensuremath{\xi}}^{\ensuremath{-}z}\ensuremathξ\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.
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.
Recently, the driven dynamics of localization phase transition have garnered growing interest. However, studies so far have mainly considered initial localized states, whose driven dynamics follow the Kibble-Zurek mechanism (KZM). In this study, we investigate the driven dynamics of the localization phase transition in the Aubry-André model starting from a gapless extended state, which violates the adiabatic-impulse scenario of KZM. By linearly driving the quasiperiodic potential strength across the critical point, we numerically simulate the driven dynamics and analyze the scaling behavior of both the inverse participation ratio ( I ) and the dynamic deviation from the instantaneous ground-state energy ( D ) . We demonstrate that the driven dynamics starting from initially extended states satisfies the criterion for the applicability of KZM and its extension, finite-time scaling (FTS). The scaling functions governing the driven dynamics of both I and D have been derived based on FTS and numerically validated. We found that the scaling functions exhibit significant differences at large R and small R , and also differ considerably from the scaling functions when the initial state is localized, highlighting the crucial role of initial-state behavior. The established scaling laws remain robust across a wide range of system sizes and driving rates, providing testable predictions for experimental realizations.
In this work, we explore the driven dynamics of the one-dimensional localization transitions. By linearly changing the strength of disorder potential, we calculate the evolution of the localization length $\ensuremath{\xi}$ and the inverse participation ratio I\mathcal{I}I in a disordered Aubry-Andr\'e (AA) model, and investigate the dependence of these quantities on the driving rate for a single typical realization of the disorder configuration. At first, we focus on the limit in the absence of the quasiperiodic potential. We find that the driven dynamics from both ground state and excited state can be described by the Kibble-Zurek scaling (KZS). Then, the driven dynamics near the critical point of the AA model is studied. Here, since both the disorder and the quasiperiodic potential are relevant directions, the KZS should include both scaling variables. Our present work not only extends our understanding of the localization transitions but also generalizes the application of the KZS.
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.
We study a one-dimensional lattice model with site-dependent nearest-neighbor hopping amplitudes that follow a power-law profile. The hopping variation is controlled by a grading exponent, ∣alpha∣|alpha|∣alpha∣, which serves as the tuning parameter of the system. In the thermodynamic limit, the ground state becomes localized in the limit ∣alpha∣→0|alpha| \to 0∣alpha∣→0, signaling the presence of a critical point characterized by a diverging localization length. Using exact diagonalization methods, we perform finite-size scaling analysis, and extract the associated critical exponent governing the near-critical behavior. To further characterize the criticality, we analyze inverse participation ratio (IPR), energy gap between the ground and first excited state, and fidelity-susceptibility. We also investigate the nonequilibrium dynamics by linearly ramping the hopping profile at various rates and tracking the evolution of the localization length and the IPR. The Kibble-Zurek mechanism successfully explains the resulting dynamics of the system via the critical exponents obtained from static scaling analysis. Beyond its fundamental significance, the kinetic-grading-induced localization transition provides a natural platform for quantum sensing. Using the critical enhancement of the quantum Fisher information (QFI), we demonstrate that the system enables quantum-enhanced parameter estimation of the grading exponent. We propose both adiabatic and dynamical quantum critical sensors and demonstrate that they exhibit enhanced scaling of the QFI. Our results therefore establish graded kinetic systems not only as a new setting for localization physics, but also as a potential resource for designing quantum-enhanced sensing devices.
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 $\ensuremath{\zeta}$ 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{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 ${F}_{Q}\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, 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.