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arXiv:2004.08706 · Journal of Statistical Mechanics Theory and Experiment

Entanglement revivals as a probe of scrambling in finite quantum systems

Ranjan Modak, Vincenzo Alba, Pasquale Calabrese

Published Aug 1, 2020• 67 citations
arXivHTMLPDF

Abstract

Abstract The entanglement evolution after a quantum quench became one of the tools to distinguish integrable versus chaotic (non-integrable) quantum many-body dynamics. Following this line of thoughts, here we propose that the revivals in the entanglement entropy provide a finite-size diagnostic benchmark for the purpose. Indeed, integrable models display periodic revivals manifested in a dip in the block entanglement entropy in a finite system. On the other hand, in chaotic systems, initial correlations get dispersed in the global degrees of freedom (information scrambling) and such a dip is suppressed. We show that while for integrable systems the height of the dip of the entanglement of an interval of fixed length decays as a power law with the total system size, upon breaking integrability a much faster decay is observed, signalling strong scrambling. Our results are checked by exact numerical techniques in free-fermion and free-boson theories, and by time-dependent density matrix renormalisation group in interacting integrable and chaotic models.

DOI
10.1088/1742-5468/aba9d9

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Abstract The entanglement evolution after a quantum quench became one of the tools to distinguish integrable versus chaotic (non-integrable) quantum many-body dynamics. Following this line of thoughts, here we propose that the revivals in the entanglement entropy provide a finite-size diagnostic benchmark for the purpose. Indeed, integrable models display periodic revivals manifested in a dip in the block entanglement entropy in a finite system. On the other hand, in chaotic systems, initial correlations get dispersed in the global degrees of freedom (information scrambling) and such a dip is suppressed. We show that while for integrable systems the height of the dip of the entanglement of an interval of fixed length decays as a power law with the total system size, upon breaking integrability a much faster decay is observed, signalling strong scrambling. Our results are checked by exact numerical techniques in free-fermion and free-boson theories, and by time-dependent density matrix renormalisation group in interacting integrable and chaotic models.
How quantum information is scrambled in the global degrees of freedom of nonequilibrium many-body systems is a key question to understand local thermalization. A consequence of scrambling is that in the scaling limit the mutual information between two intervals vanishes at all times, i.e., it does not exhibit a peak at intermediate times. Here we investigate the mutual information scrambling after a quantum quench in both integrable and nonintegrable one-dimensional systems. We study the mutual information between two intervals of finite length as a function of their distance. In integrable systems, the mutual information exhibits an algebraic decay with the distance between the intervals, signaling weak scrambling. This behavior may be qualitatively understood within the quasiparticle picture for the entanglement spreading. In the scaling limit of large intervals, times, and distances between the intervals, with their ratios fixed, this predicts a decay exponent equal to 1/21/21/2. Away from the scaling limit, the power-law behavior persists, but with a larger (and model-dependent) exponent. For nonintegrable models, a much faster decay is observed, which can be attributed to the finite lifetime of the quasiparticles: unsurprisingly, nonintegrable models are better scramblers.
Investigating the scrambling of quantum information (also known as quantum chaos) in many-body quantum systems provides valuable insights into the postquench evolution of a system. While various methods are available to understand the chaotic dynamics of these quantum systems, scrambling of mutual information following a quench has recently also emerged as a diagnostic of chaotic dynamics. In this work, we consider a chain of spin-1/21/21/2 particles of a finite length LLL evolving with the mixed-field Ising Hamiltonian and impose open boundary conditions. We simulate the time evolution of entanglement entropy and mutual information following a quench from the N\'eel state in this system using tensor networks and find that the entanglement entropy for nonintegrable systems saturates to a constant value at late times, while it continues to oscillate for integrable systems. We also find that mutual information peaks as a function of distance between intervals decay faster for nonintegrable systems compared to integrable systems, in agreement with the results in literature for XXZ chains. We compare the oscillations in entanglement entropy evolution obtained from simulations in the integrable case with analytic results from the quasiparticle picture and find agreement.
The dynamics of a quantum system encode signatures of whether the underlying Hamiltonian is integrable or chaotic, giving rise to the concept of quantum information scrambling through the properties of the resulting dynamical states or operators. We introduce an information-theoretic framework based on the Haar-averaged sum of total correlations (aSTC), together with average genuine multipartite entanglement generated dynamically from initially fully separable states, as robust probes of quantum information scrambling. Using the long-range quantum XYZ spin model in transverse and longitudinal magnetic fields, whose integrable limit is the nearest-neighbor transverse XY model, we demonstrate that the long-time average and, more importantly, the temporal fluctuations of the aSTC provide a faithful and system-size-independent signature of integrable and chaotic dynamics, similar to the conventional measure of scrambling, out-of-time-ordered correlator (OTOC). When the system is in contact with the thermal reservoir and system-bath coupling follows Markovianity, we find that the fluctuations of the aSTC and OTOC continue to distinguish integrable and chaotic dynamics only at intermediate times. However, we observe that in the non-Markovian domain, information backflow restores the scrambling dynamics, enabling the aSTC to retain its distinguishing power even at long times. Interestingly, we exhibit that, under Markovian amplitude damping and non-Markovian dephasing noise, the temporal fluctuations of the aSTC can discriminate between integrability and non-integrability in the weak Markovian regime, even when OTOC fails to do so.
The linear growth of entanglement after a quench from a state with short-range correlations is a universal feature of many body dynamics. It has been shown to occur in integrable and chaotic systems undergoing either Hamiltonian, Floquet or circuit dynamics and has also been observed in experiments. The entanglement dynamics emerging from long-range correlated states is far less studied, although no less viable using modern quantum simulation experiments. In this work, we investigate the dynamics of the bipartite entanglement entropy and mutual information from initial states which have long-range entanglement with correlation between antipodal points of a finite and periodic system. Starting from these crosscap states, we study both brickwork quantum circuits and Hamiltonian dynamics and find distinct patterns of behaviour depending on the type of dynamics and whether the system is integrable or chaotic. Specifically, we study both dual unitary and random unitary quantum circuits as well as free and interacting fermion Hamiltonians. For integrable systems, we find that after a time delay the entanglement experiences a linear in time decrease followed by a series of revivals, while, in contrast, chaotic systems exhibit constant entanglement entropy. On the other hand, both types of systems experience an immediate linear decrease of the mutual information in time. In chaotic systems this then vanishes, whereas integrable systems instead experience a series of revivals. We show how the quasiparticle and membrane pictures of entanglement dynamics can be modified to describe this behaviour, and derive explicitly the quasiparticle picture in the case of free fermion models which we then extend to all integrable systems.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.
A pure quantum state can fully describe thermal equilibrium as long as one focuses on local observables. The thermodynamic entropy can also be recovered as the entanglement entropy of small subsystems. When the size of the subsystem increases, however, quantum correlations break the correspondence and mandate a correction to this simple volume law. The elucidation of the size dependence of the entanglement entropy is thus essentially important in linking quantum physics with thermodynamics. Here we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing equilibrium. We numerically find that our formula applies universally to any sufficiently scrambled pure state representing thermal equilibrium, i.e., energy eigenstates of non-integrable models and states after quantum quenches. Our formula is exploited as diagnostics for chaotic systems; it can distinguish integrable models from non-integrable models and many-body localization phases from chaotic phases.