Quasiconservation laws and suppressed transport in weakly interacting localized models

The stability of localization in the presence of interactions remains an open problem, with finite-size effects posing significant challenges to numerical studies. In this work, we investigate the perturbative stability of noninteracting localization under weak interactions, which allows us to analyze much larger system sizes. Focusing on disordered Anderson and quasiperiodic Aubry-Andr\'e models in one dimension, and using the adiabatic gauge potential at first order in perturbation theory, we compute first-order corrections to noninteracting local integrals of motion (LIOMs). We find that for at least an $O(1)$ fraction of the LIOMs, the corrections are well-controlled and converge at large system sizes, while others suffer from resonances. Additionally, we introduce and study the charge-transport capacity of this weakly interacting model. To first order, we find that the charge transport capacity remains bounded in the presence of interactions. Taken together, these results demonstrate that localization is perturbatively stable to weak interactions at first order, implying that, at the very least, localization persists for parametrically long times in the inverse interaction strength. We expect this perturbative stability to extend to all orders at sufficiently strong disorder, where the localization length is short, representing the true localized phase. Conversely, our findings suggest that the previously proposed interaction-induced avalanche instability, namely, in the weakly localized regime of the Anderson and Aubry-Andr\'e models, is a more subtle phenomenon arising only at higher orders in perturbation theory or through nonperturbative effects.