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arXiv:2301.00155 · Physical Review A

Kibble-Zurek scaling in one-dimensional localization transitions

Xuan Bu, Liang-Jun Zhai, Shuai Yin

Published Aug 21, 2023• 11 citations
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Abstract

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 $\mathcal{I}$ 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.

DOI
10.1103/physreva.108.023312

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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 \ensuremathξ\ensuremath{\xi}\ensuremathξ 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.
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.
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.
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 study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr{h}_{c}^{\mathrm{lsr}}hclsr​ which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.
In this work, we focus on the critical dynamics of the one-dimensional quasiperiodic p-wave-paired Aubry-André-Harper model, which exhibits a transition point between the gapped critical and the gapless localized phases. First, we disclose that the dynamical exponent features two distinct plateaus in the gapless localized phase. Besides the plateau with dynamical exponent z=1.388z=1.388z=1.388 near the transition point, there is a second one with z=1z=1z=1 away from the transition point. Then we demonstrate that these two plateaus intrinsically affect the critical dynamics with moderate quench rate by employing both one-way and round-trip quench protocols. In the one-way quench protocol, we clarify that the Kibble-Zurek (KZ) regime consists of two sub-regimes with different KZ exponents, which is a direct consequence of the two-plateau structure of the dynamical exponent. We also diagnose the KZ, pre-saturated, and saturated regimes by varying the quench rate from slow to fast limits. While, in the round-trip quench protocol, we confirm the interference effect of two critical dynamics and show a narrow neck in the oscillatory density of defects. It turns out that the position of the neck depends on the ratio of to-and-fro quench rates and the p-wave superconductivity pairing amplitude. We show how the position of the neck can be used to quantitatively determine the turning point of the two KZ sub-regimes.