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arXiv:2302.09793 · Physical Review A

Scaling laws of the out-of-time-order correlators at the transition to the spontaneous PT -symmetry breaking in a Floquet system

Wen-Lei Zhao, Ru-Ru Wang, Han Ke, Jie Liu

Published Jun 1, 2023• 15 citations
arXivHTMLPDF

Abstract

We investigate both numerically and analytically the dynamics of out-of-time-order correlators (OTOCs) in a non-Hermitian kicked rotor model, addressing the scaling laws of the time dependence of OTOCs at the transition to the spontaneous $\mathcal{PT}$-symmetry breaking. In the unbroken phase of $\mathcal{PT}$ symmetry, the OTOCs increase monotonically and eventually saturate with time, demonstrating the freezing of information scrambling. Just beyond the phase transition points, the OTOCs increase in the power laws of time, with the exponent being larger than 2. Interestingly, the quadratic growth of OTOCs with time emerges when the system is far beyond the phase transition points. The above numerical findings are validated by our theoretical analysis, which provides a general framework with important implications for Floquet engineering and information scrambling in chaotic systems.

DOI
10.1103/physreva.107.062201

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We investigate both numerically and analytically the dynamics of out-of-time-order correlators (OTOCs) in a non-Hermitian kicked rotor model, addressing the scaling laws of the time dependence of OTOCs at the transition to the spontaneous PT\mathcal{PT}PT-symmetry breaking. In the unbroken phase of PT\mathcal{PT}PT symmetry, the OTOCs increase monotonically and eventually saturate with time, demonstrating the freezing of information scrambling. Just beyond the phase transition points, the OTOCs increase in the power laws of time, with the exponent being larger than 2. Interestingly, the quadratic growth of OTOCs with time emerges when the system is far beyond the phase transition points. The above numerical findings are validated by our theoretical analysis, which provides a general framework with important implications for Floquet engineering and information scrambling in chaotic systems.
We investigate the dynamics of the out-of-time-order correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies PT -symmetry. The spontaneous PT -symmetry breaking emerges when the strength of the imaginary part of the kicking potential exceeds a threshold value. We find, both analytically and numerically, that in the broken phase of PT symmetry, the OTOCs rapidly saturate with time evolution. Interestingly, the late-time saturation value scales as a pow-law in the system size. The mechanism of such scaling law results from the interplay between the effects of the nonlocal operator in OTOCs and the time reversal induced by non-Hermitian-driven potential.
We investigate the dynamics of the out-of-time-ordered correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies PT\mathcal{PT}PT-symmetry. The spontaneous PT\cal{PT}PT-symmetry breaking emerges when the strength of the imaginary part of the kicking potential exceeds a threshold value. We find, both analytically and numerically, that in the broken phase of PT\cal{PT}PT symmetry, the OTOCs rapidly saturate with time evolution. Interestingly, the late-time saturation value scales as a pow-law in the system size. The mechanism of such scaling law results from the interplay between the effects of nonlocal operator in OTOCs and the time reversal induced by non-Hermitian driven potential.
We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.
We investigate both analytically and numerically the dynamics of quantum scrambling, characterized by out-of-time ordered correlators (OTOCs), in a non-Hermitian quantum kicked rotor subject to quantum resonance conditions. Analytical expressions for OTOCs as a function of time are obtained, demonstrating a sudden transition from linear growth to quadratic growth when the non-Hermitian parameter decays to zero. At this critical point, the rates of the linear growth are found to diverge to infinity, indicating the existence of quantum criticality at the boundary of the non-Hermitian regime. The underlying mechanism of this quantum criticality is uncovered, and possible applications in quantum metrology are discussed.
This paper reports the findings of the late time behavior of the out-of-time-ordered correlators (OTOCs) via quantum kicked rotor and kicked top models with PT\mathcal{PT}PT-symmetric driving potential. An analytical expression of the OTOCs' power-law growth with time is yielded as $C(t)=G(K){t}^{\ensuremath{\gamma}}$, with $\ensuremath{\gamma}$ being system dependent. Interestingly, the growth rate GGG features a quantized response to the increase of the kick strength KKK. The physics behind this is the quantized absorption of energy from the non-Hermitian driving potential. This discovery and the ensuing establishment of the quantization mechanism in the dynamics of quantum chaos with non-Hermiticity will provide insights in chaotic dynamics, promising unprecedented observations in updated experiments.
We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify "information scrambling" due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system's dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival's conjecture to partition the eigenstates of the Floquet map into "regular" and "chaotic." Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.