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In this paper we study one-dimensional quantum Ising spin chains in externalmagnetic field close to an integrable point. We concentrate on the dynamics ofthe slowest operator, that plays a key role at the final period ofthermalization. We introduce two independent definitions of the slowestoperator: local and translationally-invariant ones. We construct both operatorsnumerically using tensor networks and extensively compare their physicalproperties. We find that the local operator has a significant overlap withenergy flux, it does not correspond to an integral of motion, and, as one goesaway from the integrable point, its revivals get suppressed and the rate ofdelocalization changes from extremely slow to slower than diffusion. Thetranslationally-invariant operator corresponds to an integral of motion; as thesystem becomes less integrable, at some point this operator changes its nature:from no overlap with any magnetization and fast rate of delocalization, tonon-zero overlap with magnetizations $\sigma_{x}$ and $\sigma_{z}$ and slowrate of delocalization.

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