Long-Time Limits of Local Operator Entanglement in Interacting Integrable Models

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 [PRL $\textbf{122}$, 250603; Commun. Math. Phys. $\textbf{371}$, 651-688] showing LOE grows logarithmically in the early time regime, $t\ll L$, to the late time regime, $t\gg L $. 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.