More on the Operator Space Entanglement (OSE): Rényi OSE, revivals, and integrability breaking

We investigate the dynamics of the Rényi 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 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 (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.