Finite-size generators for weak integrability breaking perturbations in the Heisenberg chain

An integrable model perturbed by special ``weak-integrability-breaking'' perturbations thermalizes at timescales much longer than predicted by Fermi's golden rule. Recently, a systematic construction of such perturbations based on the so-called long-range deformations of integrable chains was formulated. These perturbations, obtained as truncations of the long-range deformations in some small parameter expansions, can be viewed as produced by unitary rotations of the short-range integrable models. For infinite systems, several ``generators'' (extensive local, boosted, and bilocal operators) of weak perturbations are known. The main aim of this work is to understand the appropriate generators in finite systems with periodic boundaries since simple counterparts to boosted and bilocal operators are not known in such cases. We approach this by studying the structure of the adiabatic gauge potential (AGP), a proxy for such generators in finite chains, which was originally introduced as a sensitive measure of quantum chaos. We prove an exact relation between the AGPs for the boosted and bilocal classes of generators and note that the counterpart to the boost does not appear in a closed analytic form in finite systems but shows quasilocality nonetheless. We also introduce and study strictly local variants of weak-integrability-breaking perturbations.