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arXiv:1804.01545 · Physical Review A

Signature of chaos and delocalization in a periodically driven many-body system: An out-of-time-order-correlation study

Sayak Ray, Subhasis Sinha, K. Sengupta

Published Nov 27, 2018• 35 citations
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Abstract

We study out-of-time-order correlation (OTOC) for one-dimensional periodically driven hardcore bosons in the presence of Aubry-Andr\'e (AA) potential and show that both the spectral properties and the saturation values of OTOC in the steady state of these driven systems provide a clear distinction between the localized and delocalized phases of these models. Our results, obtained via exact numerical diagonalization of these boson chains, thus indicate that OTOC can provide a signature of drive-induced delocalization, even for systems which do not have a well-defined semiclassical (and/or large $N$) limit. We demonstrate the presence of such a signature by analyzing two different drive protocols for hardcore boson chains leading to distinct physical phenomena and discuss experiments which can test our theory.

DOI
10.1103/physreva.98.053631

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We study out-of-time-order correlation (OTOC) for one-dimensional periodically driven hardcore bosons in the presence of Aubry-Andr'e (AA) potential and show that both the spectral properties and the saturation values of OTOC in the steady state of these driven systems provide a clear distinction between the localized and delocalized phases of these models. Our results, obtained via exact numerical diagonalization of these boson chains, thus indicate that OTOC can provide a signature of drive-induced delocalization, even for systems which do not have a well-defined semiclassical (and/or large NNN) limit. We demonstrate the presence of such a signature by analyzing two different drive protocols for hardcore boson chains leading to distinct physical phenomena and discuss experiments which can test our theory.
Motivated by the recent experiment by Bordia et al. [Nat. Phys. 13, 460 (2017)NPAHAX1745-247310.1038/nphys4020], we study the single particle delocalization phenomena of the Aubry-André (AA) model subjected to periodic drives. In two distinct cases we construct an equivalent classical description to illustrate that the drive-induced delocalization phenomena stems from an instability and the onset of chaos in the underlying dynamics. In the first case we analyze the delocalization and the thermalization in a time modulated AA potential with respect to driving frequency and demonstrate that there exists a threshold value of the amplitude of the drive. In the next example, we show that the periodic modulation of the phase of the hopping amplitude induced by a gauge field leads to an unusual effect on delocalization with a nonmonotonic dependence on the driving frequency. Within a window of such a driving frequency a delocalized Floquet band with a mobility edge appears, exhibiting multifractality in the spectrum as well as in the Floquet eigenfunctions. Finally, we explore the effect of interaction and discuss how the results of the present analysis can be tested experimentally.
We study the localization-delocalization transition of Floquet eigenstates in a driven fermionic chain with an incommensurate Aubry-Andr\'e potential and a hopping amplitude which is varied periodically in time. Our analysis shows the presence of a mobility edge separating single-particle delocalized states from localized and multifractal states in the Floquet spectrum. Such a mobility edge does not have any counterpart in the static Aubry-Andr\'e model and exists for a range of drive frequencies near the critical frequency at which the transition occurs. The presence of the mobility edge is shown to leave a distinct imprint on fermion transport in the driven chain; it also influences the Shannon entropy and the survival probability of the fermions at long times. In addition, we find the presence of CAT (linear superposition of localized single particle states) states in the Floquet spectrum with weights centered around a few nearby sites of the chain. This is shown to be tied to the flattening of Floquet bands over a range of quasienergies. We support our numerical studies with a semianalytic expression for the Floquet Hamiltonian (HF{H}_{F}HF​) computed within a Floquet perturbation theory. The eigenspectra of the perturbative HF{H}_{F}HF​ so obtained exhibit qualitatively identical properties to the exact eigenstates of HF{H}_{F}HF​ obtained numerically. Our results thus constitute an analytic expression of a HF{H}_{F}HF​ whose spectrum supports multifractal and cat states. We suggest experiments which can test our theory.
We study the many-body localization (MBL) transition of Floquet eigenstates in a driven, interacting fermionic chain with an incommensurate Aubry-Andr\'e potential and a time-periodic hopping amplitude as a function of the drive frequency ${\ensuremath{\omega}}_{D}$ using exact diagonalization (ED). We find that the nature of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing frequency; moreover, for a significant range of intermediate ${\ensuremath{\omega}}_{D}$, the Floquet eigenstates exhibit nontrivial fractal dimensions. We find a possible transition from the ergodic to this multifractal phase followed by a gradual crossover to the MBL phase as the drive frequency is increased. We also study the fermion autocorrelation function, entanglement entropy, normalized participation ratio (NPR), fermion transport, and the inverse participation ratio (IPR) as a function of ${\ensuremath{\omega}}_{D}$. We show that the autocorrelation, fermion transport, and NPR display qualitatively different characteristics (compared to their behavior in the ergodic and MBL regions) for the range of ${\ensuremath{\omega}}_{D}$ which supports multifractal eigenstates. In contrast, the entanglement growth in this frequency range tend to have similar features as in the MBL regime; its rate of growth is controlled by ${\ensuremath{\omega}}_{D}$. Our analysis thus indicates that the multifractal nature of Floquet-MBL eigenstates can be detected by studying autocorrelation function and fermionic transport of these driven chains. We support our numerical results with a semianalytic expression of the Floquet Hamiltonian obtained using Floquet perturbation theory (FPT) and discuss possible experiments which can test our predictions.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/21/21/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t)(1/t)(1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1)(1/t^α; α\ge 1)(1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.
Abstract The dynamic region of out-of-time-ordered correlators (OTOCs) serves as a powerful indicator of chaos in classical and semiclassical systems, capturing the characteristic exponential growth. However, in spin systems, the dynamic region of OTOCs does not reliably quantify chaos as both integrable and chaotic systems display similar behavior. Instead, we utilize the saturation behavior of OTOCs to differentiate between chaotic and integrable regimes. In integrable systems, the saturation region of OTOCs exhibits oscillatory behavior, while in chaotic systems, it shows a stable saturation. To evaluate this distinction, we investigate a time-dependent Ising spin system subjected to a linearly ramping transverse field, analyzing both integrable (without longitudinal field) and non-integrable (with longitudinal field) scenarios. This setup accelerates system ergodicity due to the introduction of an additional time scale within the system, enhancing chaotic dynamics in the non-integrable regime, and provides a compelling model for studying the interplay between integrability and chaos in quantum systems. To further support our findings, we investigate the level spacing distribution of time-dependent unitary operators, which effectively distinguishes chaotic from regular regions in our system and corroborates the results obtained from the saturation behavior of the OTOC. Additionally, we calculate a metric for the normalized Fourier spectrum of the OTOC which is dependent on the number of frequency components present to gain insights into the observed oscillations and its dependence on the ramping field. Graphical abstract
We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. The MBL phase exhibits logarithmic in time growth of entanglement entropy when the system is initially prepared in a product state, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion, which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggest a direct transition between the two phases.