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

TL;DR

This paper numerically constructs and analyzes translationally invariant quasi-conserved operators in a non-integrable quantum spin chain, revealing their decay properties, localization, and role in prethermalization.

Contribution

The authors develop a numerical method to construct quasi-conserved operators with maximum range up to 12, linking them to dressed total spin-z operators and analyzing their properties.

Findings

Residual commutator norm decays exponentially with range at small M

Quasi-conserved operators are localized in operator Hilbert space

Prethermalization dynamics are described by a generalized Gibbs ensemble including these operators

Abstract

We numerically construct translationally invariant quasi-conserved operators with maximum range M which best-commute with a non-integrable quantum spin chain Hamiltonian, up to M = 12. In the large coupling limit, we find that the residual norm of the commutator of the quasi-conserved operator decays exponentially with its maximum range M at small M, and turns into a slower decay at larger M. This quasi-conserved 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 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 quasi-conserved operator.