Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalization in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analyzing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behavior. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic-driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behavior in quantum systems, beyond the initial exponential growth in OTOCs.
Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalization in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analyzing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behavior. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic-driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behavior in quantum systems, beyond the initial exponential growth in OTOCs.
A bstract In semi-classical systems, the exponential growth of the out-of-time-order correlator (OTOC) is believed to be the hallmark of quantum chaos. However, on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle-dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.
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-1/2 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) relaxation for integrable systems and a higher-degree power-law decay (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.
Fast scrambling of quantum correlations, reflected by the exponential growth of out-of-time-order correlators (OTOCs) on short pre-Ehrenfest time scales, is commonly considered as a major quantum signature of unstable dynamics in quantum systems with a classical limit. In two recent works [Phys. Rev. Lett. 123, 160401 (2019)0031-900710.1103/PhysRevLett.123.160401] and [Phys. Rev. Lett. 124, 140602 (2020)10.1103/PhysRevLett.124.140602], a significant difference in the scrambling rate of integrable (many-body) systems was observed, depending on the initial state being semiclassically localized around unstable fixed points or fully delocalized (infinite temperature). Specifically, the quantum Lyapunov exponent λ_{q} quantifying the OTOC growth is given, respectively, by λ_{q}=2λ_{s} or λ_{q}=λ_{s} in terms of the stability exponent λ_{s} of the hyperbolic fixed point. Here we show that a wave packet, initially localized around this fixed point, features a distinct dynamical transition between these two regions. We present an analytical semiclassical approach providing a physical picture of this phenomenon, and support our findings by extensive numerical simulations in the whole parameter range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest that the existence of this crossover is a hallmark of unstable separatrix dynamics in integrable systems, thus opening the possibility to distinguish the latter, on the basis of this particular observable, from genuine chaotic dynamics generally featuring uniform exponential growth of the OTOC.
Abstract In this article, we propose higher-point out-of-time-order correlators (OTOCs) as a tool to differentiate chaotic from saddle-dominated dynamics in late times. As a model, we study the scrambling dynamics in supersymmetric quantum mechanical systems. Using the eigenstate representation, we define the 2 N -point OTOC using two formalisms, namely the ’Tensor Product formalism’ and the ’Partner Hamiltonian formalism’. We analytically find that the 2 N -point OTOC for the supersymmetric 1D harmonic oscillator is in exact agreement with that of the 1D bosonic harmonic oscillator system. We show that the higher-point OTOC is a more sensitive measure of scrambling than the usual 4-point OTOC. To demonstrate this, we analyze a supersymmetric sextic 1D oscillator, for which the bosonic partner system has an unstable saddle in the phase space, while the saddle is absent in the fermionic counterpart. For such a system, we show that the saddle-dominated scrambling, higher anharmonic potential effects, and the supersymmetric OTOC exhibit similar dynamics due to supersymmetry constraints. Finally, we illustrate that the late-time dynamics of the higher-point OTOC become oscillatory after the peak for saddle-dominated scrambling and anharmonic oscillator systems. We propose the higher-point OTOC as a probe of late-time dynamics in non-chaotic systems that exhibit fast early-time scrambling.
Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.
This PhD thesis investigates scrambling in near-integrable and chaotic systems. First, we present a quasiclassical formalism of out-of-time-ordered correlators (OTOCs), the main chaos probe that we use, using the semiclassical van Vleck-Gutzwiller propagator and the application of the diagonal approximation. For short evolution times, this quasiclassical approach yields the same result as the Wigner-Moyal formalism, i.e. an exponential growth in a chaotic regime. For long times, for which the semiclassical framework is, in principle, still valid, the diagonal approximation yields an asymptotic saturation value for the quasiclassical OTOC under the assumption of fully developed classical chaos. Numerical simulations in Bose-Hubbard systems show that this estimation strongly underestimates the actual threshold value of the quantum OTOCs. This indicates that genuinely quantum contributions are primarily responsible for describing OTOCs beyond the short-time regime. The thesis then shifts its focus to near-integrable systems, where chaos is present only in the form of a thin chaotic layer. In a fully integrable system, around a fixed point, the dynamics is dictated by a stability exponent, and in a fully chaotic system by a maximal Lyapunov exponent. The question remains open as to how the two compete in near-integrable regimes. We describe this integrability-chaos transition by establishing a description of the dynamics inside the chaotic layer in a universal system: the periodically driven pendulum. We support our claim for the universality of this mechanism by studying two paradigmatic examples of the integrability-to-chaos transition, namely the kicked rotor and the kicked Bose-Hubbard dimer. Finally, we study quantum scars in single- and many-particle systems. For the former, we find that the short-time behaviour of indicators such as the OTOC, computed in the kicked rotor, is not sensitive to the presence of a scar, i.e. to the fulfilment of Heller's criterion. Nonetheless, we show that the OTOC can serve as a probe of unstable periodic orbits. For the latter, we look at the persistent oscillations that manifest in the presence of scars in different observables computed for staggered dimer states in Bose-Hubbard models. As the interaction strength increases, one expects scarring to be a less prevalent phenomenon. This is corroborated by the oscillations that progressively vanish.