More on the Operator Space Entanglement (OSE): R\'enyi OSE, revivals, and integrability breaking

We investigate the dynamics of the R\'enyi Operator Space Entanglement ($OSE$) entropies $S_n$ 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 R\'enyi $OSE$ entropies of diagonal operators with nonzero trace saturate at long times, in contrast with the behavior of von Neumann entropy. Oppositely, the R\'enyi 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 ($ES$) 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, $S_n$ exhibit strong revivals, which are washed out when integrability is broken.