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arXiv:2411.13469 · Physical review. B./Physical review. B

Information scrambling and entanglement dynamics in Floquet time crystals

Himanshu Sahu, Fernando Iemini

Published Mar 5, 2025• 1 citation
arXivHTMLPDF

Abstract

We study the dynamics of out-of-time-ordered correlators (OTOCs) and the entanglement of entropy as quantitative measures of information propagation in disordered many-body systems exhibiting Floquet time-crystal (FTC) phases. We find that an OTOC spreads in the FTC with different characteristic timescales due to the existence of a preferred ``quasiprotected'' direction, denoted as the $\ensuremath{\ell}$-bit direction, along which the spins stabilize their period-doubling magnetization for exponentially long times. While orthogonal to this direction the OTOC thermalizes as an usual many-body localization time-independent system (at stroboscopic times), along the $\ensuremath{\ell}$-bit direction the system features a more complex structure. The scrambling appears as a combination of an initially frozen dynamics (while in the stable period-doubling magnetization time window) and a later logarithmic slow growth (over its decoherence regime) until full thermalization. Interestingly, in the late-time regime, since the wave front propagation of correlations has already settled through the whole chain, scrambling occurs at the same rate regardless of the distance between the spins, thus resulting in an overall envelopelike structure of all OTOCs, independent of their distance, merging into a single growth. In addition, the entanglement entropy shows a logarithmic growth over all time, reflecting the slow dynamics up to a thermal volume-law saturation.

DOI
10.1103/physrevb.111.104302

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We study the dynamics of out-of-time-ordered correlators (OTOCs) and the entanglement of entropy as quantitative measures of information propagation in disordered many-body systems exhibiting Floquet time-crystal (FTC) phases. We find that an OTOC spreads in the FTC with different characteristic timescales due to the existence of a preferred ``quasiprotected'' direction, denoted as the \ensuremathℓ\ensuremath{\ell}\ensuremathℓ-bit direction, along which the spins stabilize their period-doubling magnetization for exponentially long times. While orthogonal to this direction the OTOC thermalizes as an usual many-body localization time-independent system (at stroboscopic times), along the \ensuremathℓ\ensuremath{\ell}\ensuremathℓ-bit direction the system features a more complex structure. The scrambling appears as a combination of an initially frozen dynamics (while in the stable period-doubling magnetization time window) and a later logarithmic slow growth (over its decoherence regime) until full thermalization. Interestingly, in the late-time regime, since the wave front propagation of correlations has already settled through the whole chain, scrambling occurs at the same rate regardless of the distance between the spins, thus resulting in an overall envelopelike structure of all OTOCs, independent of their distance, merging into a single growth. In addition, the entanglement entropy shows a logarithmic growth over all time, reflecting the slow dynamics up to a thermal volume-law saturation.
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.
The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, N Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For t≲sqrt[N], after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At t∼sqrt[N], the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading 1/N effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at t∼2N, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at t≲sqrt[N] indicate lack of thermalization-a desired feature for better performance of quantum information devices.
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.
Current quantum hardware is limited by noise and decoherence, which restrict the depth of unitary circuits that can be implemented with high fidelity. We investigate how the compressibility of time-evolution operators depends on the dynamical regime of the underlying many-body system. As a testbed, we study a one-dimensional Floquet random circuit with a tunable competition between interactions and on-site disorder. Using tensor-network simulations, we characterize operator growth through the operator-entanglement entropy of the Floquet unitary as well as of out-of-time-ordered correlators (OTOCs). We find rapid operator scrambling at weak disorder, while strong disorder leads to slow OTOC-front propagation and logarithmic or near-logarithmic operator-entanglement growth over the accessible time window. We then optimize shallow brickwall circuits to approximate the Floquet evolution and show that strong-disorder circuits can be compressed to substantially smaller depths than weak-disorder circuits at fixed logarithmic fidelity density. These results suggest that localized or slowly scrambling dynamics provide a favorable regime for compressed quantum simulation on noisy devices.
Out-of-time-order correlators (OTOC) in the Ising Floquet system, which can be both integrable and nonintegrable, are studied. Instead of localized spin observables, we study contiguous symmetric blocks of spins or random operators localized on these blocks as observables. We find only power-law growth of OTOC in both integrable and nonintegrable regimes. In the nonintegrable regime, beyond the scrambling time, there is an exponential saturation of the OTOC to values consistent with random matrix theory. This motivates the use of ``prescrambled'' random block operators as observables. A pure exponential saturation of OTOC in both integrable and nonintegrable system is observed, without a scrambling phase. Averaging over random observables from the Gaussian unitary ensemble, the OTOC is found to be exactly same as the operator entanglement entropy, whose exponential saturation has been observed in previous studies of such spin chains.
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.