Physical properties of the slow operator in the Ising spin chain

We consider an Ising quantum spin chain in the neighborhood of an integrable point. The thermalization of such a system takes place in two stages, with the slow operator having a special role. We are trying to find out its physical meaning. To do this, we consider two independent definitions of a slow operator (local and translationally invariant) and construct both operators numerically using tensor networks. Then we calculate their overlap with other operators, time evolution, and OTOC. As a result, we find that the local slow operator largely coincides with the energy flow, and the translationally-invariant one changes its nature as it moves away from the integrable point. In addition, we identify the key stages of dynamics, find the speed of propagation along the chain and the transition states from integrability to non-integrability.
The report is based on arXiv:2302.07995.