Study of Adiabatic Gauge Potential for Weak Integrability Breaking Perturbations in the Heisenberg Chain

An integrable model perturbed by special "weak" perturbations thermalizes at timescales much longer than predicted by Fermi's golden rule. Recent works have proposed a systematic way of constructing families of such weak perturbations based on the so-called long-range deformations of integrable chains. The weak perturbations are obtained as truncations of the long-range deformations in some small parameter expansion and can be viewed as produced by unitary rotations of the short-range integrable models. In this study, we are primarily interested in deformations generated by boosted operators. The main aim of this work is to understand appropriate generators in systems with periodic boundary conditions since the boost is well-defined only in infinite chains. We approach this by studying the structure of the so-called adiabatic gauge potential (AGP) that was originally introduced as a very sensitive measure of quantum chaos and can serve as a proxy for such generators.