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arXiv:2108.13411 · Physical Review Letters

Out-of-Time-Ordered Crystals and Fragmentation

Berislav Buča

Published Mar 11, 2022• 27 citations
arXivHTMLPDF

Abstract

Is a spontaneous perpetual reversal of the arrow of time possible? The out-of-time-ordered correlator (OTOC) is a standard measure of irreversibility, quantum scrambling, and the arrow of time. The question may be thus formulated more precisely and conveniently: can spatially ordered perpetual OTOC oscillations exist in many-body systems? Here we give a rigorous lower bound on the amplitude of OTOC oscillations in terms of a strictly local dynamical algebra allowing for identification of systems that are out-of-time-ordered (OTO) crystals. While OTOC oscillations are possible for few-body systems, due to the spatial order requirement OTO crystals cannot be achieved by effective single or few body dynamics, e.g., a pendulum or a condensate. Rather they signal perpetual motion of quantum scrambling. It is likewise shown that if a Hamiltonian satisfies this novel algebra, it has an exponentially large number of local invariant subspaces, i.e., Hilbert space fragmentation. Crucially, the algebra, and hence the OTO crystal, are stable to local unitary and dissipative perturbations. A Creutz ladder is shown to be an OTO crystal, which thus perpetually reverses its arrow of time.

DOI
10.1103/physrevlett.128.100601

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Is a spontaneous perpetual reversal of the arrow of time possible? The out-of-time-ordered correlator (OTOC) is a standard measure of irreversibility, quantum scrambling, and the arrow of time. The question may be thus formulated more precisely and conveniently: can spatially ordered perpetual OTOC oscillations exist in many-body systems? Here we give a rigorous lower bound on the amplitude of OTOC oscillations in terms of a strictly local dynamical algebra allowing for identification of systems that are out-of-time-ordered (OTO) crystals. While OTOC oscillations are possible for few-body systems, due to the spatial order requirement OTO crystals cannot be achieved by effective single or few body dynamics, e.g., a pendulum or a condensate. Rather they signal perpetual motion of quantum scrambling. It is likewise shown that if a Hamiltonian satisfies this novel algebra, it has an exponentially large number of local invariant subspaces, i.e., Hilbert space fragmentation. Crucially, the algebra, and hence the OTO crystal, are stable to local unitary and dissipative perturbations. A Creutz ladder is shown to be an OTO crystal, which thus perpetually reverses its arrow of time.
Quantum many-body models with both Hilbert space fragmentation and non-stationarity have recently been identified. Hilbert space fragmentation does not immediately imply non-stationarity. However, strictly local dynamical symmetries directly imply non-stationarity. It is demonstrated here that these symmetries are equivalent to local fragmentation into spatially localized blocks. Using strictly local dynamical symmetries, a lower bound is given here for persistent oscillations of generalised out-of-time-ordered correlation functions (OTOCs). A novel notion of genuinely many-body continuous time translation symmetry breaking is provided by demanding non-trivial spatial modulation of the Fourier transform of the OTOC. Such non-trivial spatial-temporal dynamics stems from a perpetual backflow of quantum scrambling. Here we call systems with time-translation symmetry breaking in the OTOC, OTO crystals. This breaking cannot be realised by systems with a single effective degree of freedom (e.g. spin precession). Furthermore, the breaking is stable to all local unitary and dissipative perturbations. An XYZ Creutz ladder is presented as an example.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
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.
Out-of-time-order correlators (OTOCs) serve as a proxy for quantum information scrambling, which refers to the process where information stored locally disperses across the many-body degrees of freedom in a quantum system, rendering it inaccessible to local probes. Most experimental implementations of OTOCs to probe information scrambling rely on indirect measurements based on global observables, using techniques such as Loschmidt echoes andmultiple quantum coherences, via time-reversal evolutions. In this article, we establish a direct connection between OTOCs with global and local observables in the context of NMR experiments, where the observable is the total magnetization of the system. We conduct a numerical analysis to quantify the differences in the evolution of both magnitudes, evaluating the excitation dynamics in spin ring systems with 8 to 16 spins, using a many-body Hamiltonian and long-range interactions. Our analysis decomposes the global echo into a sum of local echoes and cross-contributions, leading to local and global OTOCs. The results indicate that, after an initial transient period, local OTOCs determine the global ones. We observe that the difference between the average of local OTOCs and the global one, as well as their fluctuations, becomes negligible as the system size increases. Thus, for large homogeneous systems, global and local OTOCs become equivalent. This behavior aligns with that observed in highly interacting or chaotic systems in several experiments.
Out-of-time-order correlators (OTOCs), recently at the center of discussion on quantum chaos, are tools to understand the information scrambling in different phases of quantum many-body systems. We propose a disordered ladder spin model, the XX ladder, which can be designed in a scalable cold atom setup to detect OTOCs with a sign reversal protocol alternative to existing approaches for evolution backward in time. We study both the clean and disordered XX ladder and characterize different phases (ergodic and many-body localized phases) of the model based on the decay properties of OTOCs. The emergent effective lightcone shows sublinear behavior, while the butterfly cones drastically differ from the lightcone via demonstrating superlinear behavior. Based on our results, one can observe how the information scrambling changes in the transition from well-studied one-dimensional spin models to unexplored two-dimensional spin models in a local setting.
The out-of-time-ordered correlator (OTOC) is a powerful tool for probing quantum information scrambling, a fundamental process by which local information spreads irreversibly throughout a quantum many-body system. Experimentally measuring the OTOC, however, is notoriously challenging due to the need for time-reversed evolution. Here, we present an experimental evaluation of the OTOC on a quantum computer, using three distinct protocols to address this challenge: the rewinding time method (RTM), the weak-measurement method (WMM), and the irreversibility-susceptibility method (ISM). Our experiments investigate the quantum dynamics of an XXZ spin-1/2 chain prepared in a thermal Gibbs state. As a key contribution, we provide the first experimental demonstration of the ISM, using the trapped-ion quantum computer, reimei. We also conduct a detailed comparative analysis of all three methods, revealing method-dependent behaviors in the measured OTOC. This work not only validates these protocols as practical tools for exploring quantum chaos on near-term hardware but also offers crucial insights into their respective advantages and limitations, providing a practical framework for future experimental investigations.