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arXiv:2010.07089 · Physica Scripta

Late time dynamics in SUSY saddle-dominated scrambling through higher-point OTOC

Rathindra Nath Das, Sourav Dutta, Archana Maji

Published Jul 12, 2024• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1402-4896/ad629d

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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.
In this article we study the presence of chaos in supersymmetric(SUSY) quantum mechanics. For that purpose we present a form of 4-point out of time order correlator(OTOC) for SUSY quantum mechanical systems using tensor product formalism. We calculate the 4-point OTOC for SUSY 1D harmonic oscillator and find that the OTOC is exactly equal to that of 1D bosonic harmonic oscillator system. In similar manner using the eigenstate representation of supersymmetric systems we calculate the generalized higher order out of time order correlator. The higher order OTOC is a more sensitive measure of chaos than the usual 4-point correlator used in literature. Finally, we calculate the generalized 2N-point OTOC for SUSY 1D harmonic oscillator.
The concept of out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further we demonstrate an equivalent formalism of computation using a general time independent Hamiltonian having well defined eigenstate representation for integrable supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism we consider the two well known models viz. Harmonic Oscillator and one dimensional potential well within the framework of supersymmetry. For the Harmonic Oscillator case we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both microcanonical and canonical ensembles in quantum mechanics without supersymmetry. On the other hand, for one dimensional potential well problem we found significantly different time scale and the other parameter dependence compared to the results obtained from non-supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model independent Hamiltonian along with the previously mentioned these well cited models.
Out-of-time-order correlators (OTOCs) are central probes of quantum scrambling, and their generalizations have recently become key primitives for both benchmarking quantum advantage and learning the structure of Hamiltonians. Yet their behavior has lacked a unified algorithmic interpretation. We show that higher-order OTOCs naturally fit within the framework of quantum signal processing (QSP): each OTOC(k)\mathrm{OTOC}^{(k)}OTOC(k) measures the 2k2k2k-th Fourier component of the phase distribution associated with the singular values of a spatially resolved truncated propagator. This explains the contrasting sensitivities of time-ordered correlators (TOCs) and higher-order OTOCs to causal-cone structure and to chaotic, integrable, or localized dynamics. Based on this understanding, we further generalize higher-order OTOCs by polynomial transformation of the singular values of the spatially resolved truncated propagator. The resultant signal allows us to construct frequency-selective filters, which we call \emph{OTOC spectroscopy}. This extends conventional OTOCs into a mode-resolved tool for probing scrambling and spectral structure of quantum many-body dynamics.
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-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.
We calculate the third-order out-of-time-order correlator (OTOC) of a simple harmonic oscillator with an additional quartic interaction using the second quantization method. We obtain analytic relations for the spectrum, Fock space states, and matrix elements of the coordinate, which are then used to numerically evaluate the OTOC. We observe that after the scrambling, the OTOC becomes a fluctuation around a saturation point at later times, which is associated with quantum chaotic behavior in systems that exhibit chaos. We analyze the early-time properties of the OTOC and find that in systems with sufficiently strong quartic interactions, an exponential growth curve fitting over a long time window clearly emerges in the third-order perturbation