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We numerically construct translationally invariant quasi-conserved operatorswith maximum range M which best-commute with a non-integrable quantum spinchain Hamiltonian, up to M = 12. In the large coupling limit, we find that theresidual norm of the commutator of the quasi-conserved operator decaysexponentially with its maximum range M at small M, and turns into a slowerdecay at larger M. This quasi-conserved operator can be understood as a dressedtotal "spin-z" operator, by comparing with the perturbative Schrieffer-Wolffconstruction developed to high order reaching essentially the same maximumrange. We also examine the operator inverse participation ratio of theoperator, which suggests its localization in the operator Hilbert space. Theoperator also shows almost exponentially decaying profile at short distance,while the long-distance behavior is not clear due to limitations of ournumerical calculation. Further dynamical simulation confirms that theprethermalization-equilibrated values are described by a generalized Gibbsensemble that includes such quasi-conserved operator.

Explicit construction of quasi-conserved local operator of translationally invariant non-integrable quantum spin chain in prethermalization