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arXiv:2503.09578 · SciPost Physics

Long-time limits of local operator entanglement in interacting integrable models

J. Alexander Jacoby, Sarang Gopalakrishnan

Published Jun 11, 2026• 0 citations
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Abstract

We explore the long-time behavior of Local Operator Entanglement entropy (LOE) in finite-size interacting integrable systems. For certain operators in the Rule 54 automaton, we prove that the LOE saturates to a value that is at most logarithmic in system size. This bound extends previous work [Phys. Rev. Lett. 122, 250603 (2019); Commun. Math. Phys. 371, 651 (2019)] showing LOE grows logarithmically in the early time regime, t\ll L t ≪ L , to the late time regime, $tL $. However, the late-time logarithmic bound relies on a feature of Rule 54 that does not generalize to other interacting integrable systems: namely, that there are only two types of quasiparticles, and therefore only two possible values of the phase shift between quasiparticles. We present a heuristic argument, supported by numerical evidence, that for generic interacting integrable systems (such as the Heisenberg spin chain) the saturated value of the LOE is volume-law in system size.

DOI
10.21468/scipostphys.20.6.162

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We explore the long-time behavior of Local Operator Entanglement entropy (LOE) in finite-size interacting integrable systems. For certain operators in the Rule 54 automaton, we prove that the LOE saturates to a value that is at most logarithmic in system size. This bound extends previous work [Phys. Rev. Lett. 122, 250603 (2019); Commun. Math. Phys. 371, 651 (2019)] showing LOE grows logarithmically in the early time regime, t\ll L t ≪ L , to the late time regime, tLtL tL. However, the late-time logarithmic bound relies on a feature of Rule 54 that does not generalize to other interacting integrable systems: namely, that there are only two types of quasiparticles, and therefore only two possible values of the phase shift between quasiparticles. We present a heuristic argument, supported by numerical evidence, that for generic interacting integrable systems (such as the Heisenberg spin chain) the saturated value of the LOE is volume-law in system size.
In a many-body quantum system, local operators in the Heisenberg picture O(t)=e^{iHt}Oe^{-iHt} spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement-the entanglement entropy in operator space-is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time t in chaotic systems, we present evidence that it grows only logarithmically in generic interacting integrable systems. Although this logarithmic growth has been previously established for noninteracting fermions, there has been no progress on interacting integrable systems to date. In this Letter we provide an analytical upper bound on operator entanglement for all local operators in the "Rule 54" qubit chain, a cellular automaton model introduced in the 1990s [Bobenko et al., CMP 158, 127 (1993)CMPHAY0010-361610.1007/BF02097234], and recently advertised as the simplest representative of interacting integrable systems. Physically, the logarithmic bound originates from the fact that the dynamics of the models is mapped onto the one of stable quasiparticles that scatter elastically. The possibility of generalizing this scenario to other interacting integrable systems is briefly discussed.
Local Operator Entanglement (LOE) has emerged an indicator of quantum chaos in many-body systems. Numerical studies have shown that, in chaotic systems, LOE grows linearly in time and displays a volume-law behavior at late times, scaling proportionally with the number of local degrees of freedom. Despite extensive numerical evidence, complemented by analytical studies in integrable systems, a fully analytical understanding of the emergence of the volume law remains incomplete. In this paper, we contribute toward this goal by deriving a late-time expression for LOE in chaotic systems that exhibits volume-law scaling. Our derivation proceeds by expressing the late-time LOE in the Liouville eigenstate basis and relies on three main assumptions: a higher-order non-resonance condition for the Hamiltonian eigenenergies, the Eigenstate Thermalization Hypothesis (ETH) ansatz for the matrix elements of the initial local operator, and the replacement of Hamiltonian eigenstates with random states in the final expression for LOE. Under these assumptions, we obtain an explicit formula displaying volume-law scaling. Finally, we complement our analytical derivation with numerical simulations of the 1D mixed-field Ising model, testing the resulting formula and exploring the regime of validity of our assumptions.
Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE ααα-Rényi entropies. Specifically, we prove that a volume law scaling for α≥1α\geq 1α≥1 implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the α<1α< 1α<1 entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.
Understanding the spreading of the operator space entanglement entropy (OSEE) is key in order to explore out-of-equilibrium quantum many-body systems. Here we argue that for integrable models the dynamics of the OSEE is related to the diffusion of the operator front. We derive the logarithmic bound 1/2ln(t)1/2ln(t)1/2ln(t) for the OSEE of some simple, i.e., low-rank, diagonal local operators. We numerically check that the bound is saturated in the rule 54 chain, which is representative of interacting integrable systems. Remarkably, the same bound is saturated in the spin-1/2 Heisenberg XXZXXZXXZ chain. Away from the isotropic point and from the free-fermion point, the OSEE grows as 1/2ln(t)1/2ln(t)1/2ln(t), irrespective of the chain anisotropy, suggesting universality. Finally, we discuss the effect of integrability breaking. We show that strong finite-time effects are present, which prevent us from probing the asymptotic behavior of the OSEE.
Abstract We investigate the dynamics of the Rényi Operator Space Entanglement (OSE) entropies Sn across several one-dimensional integrable and chaotic models. As a paradigmatic integrable system, we first consider the so-called rule 54 chain. Our numerical results reveal that the Rényi OSE entropies of diagonal operators with nonzero trace saturate at long times, in contrast with the behavior of von Neumann entropy. Oppositely, the Rényi entropies of traceless operators exhibit logarithmic growth with time, with the prefactor of this growth depending in a nontrivial manner on n. Notably, at long times, the complete operator entanglement spectrum of an operator can be reconstructed from the spectrum of its traceless part. We observe a similar pattern in the XXZ chain, suggesting Universal behavior. Additionally, we consider dynamics in nonintegrable deformations of the XXZ chain. Finite-time corrections do not allow to access the long-time behavior of the von Neumann entropy. On the other hand, for n > 1 the growth of the entropies is milder, and it is compatible with a sublinear growth, at least for operators associated with global conserved quantities. Finally, we show that in finite-size integrable systems, Sn exhibit strong revivals, which are washed out when integrability is broken.
Nonequilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling law of the bipartite entanglement entropy, thus suggesting the existence of different nonequilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes relevant in the thermodynamic sense. We show that local projective measurements induce a qualitative modification of the time growth of the entanglement entropy which changes from linear to logarithmic. However, in the stationary regime, the logarithmic behavior of the entanglement entropy does not survive in the thermodynamic limit and, for any finite value of the measurement rate, we numerically show the existence of a single area-law phase for the entanglement entropy. Finally, exploiting the quasiparticle picture, we further support our results by analyzing the fluctuations of the stationary entanglement entropy and its scaling behavior.