Explicit construction of quasiconserved local operator of translationally invariant nonintegrable quantum spin chain in prethermalization

Abstract

We numerically construct translationally invariant quasiconserved operators with maximum range $M$, which best commute with a nonintegrable quantum spin chain Hamiltonian, up to $M=12$. In the large coupling limit, we find that the residual norm of the commutator of the quasiconserved operator decays exponentially with its maximum range $M$ at small $M$, and turns into a slower decay at larger $M$. This quasiconserved operator can be understood as a dressed total ``spin-$z$'' operator, by comparing with the perturbative Schrieffer-Wolff construction developed to high order reaching essentially the same maximum range. We also examine the operator inverse participation ratio of the operator, which suggests its localization in the operator Hilbert space. The operator also shows an almost exponentially decaying profile at short distance, while the long-distance behavior is not clear due to limitations of our numerical calculation. Further dynamical simulation confirms that the prethermalization-equilibrated values are described by a generalized Gibbs ensemble that includes such quasiconserved operator.