Local vs. translationally-invariant slowest operators in quantum Ising spin chains

In this paper we study one-dimensional quantum Ising spin chains in external magnetic field close to an integrable point. We concentrate on the dynamics of the slowest operator, that plays a key role at the final period of thermalization. We introduce two independent definitions of the slowest operator: local and translationally-invariant ones. We construct both operators numerically using tensor networks and extensively compare their physical properties. We find that the local operator has a significant overlap with energy flux, it does not correspond to an integral of motion, and, as one goes away from the integrable point, its revivals get suppressed and the rate of delocalization changes from extremely slow to slower than diffusion. The translationally-invariant operator corresponds to an integral of motion; as the system becomes less integrable, at some point this operator changes its nature: from no overlap with any magnetization and fast rate of delocalization, to non-zero overlap with magnetizations $\sigma_{x}$ and $\sigma_{z}$ and slow rate of delocalization.