Robustness and independence of the eigenstates with respect to the boundary conditions across a delocalization-localization phase transition

We focus on the many-body eigenstates across a localization-delocalization phase transition. To characterize the robustness of the eigenstates, we introduce eigenstate overlaps $\mathcal{O}$ with respect to different boundary conditions. In the ergodic phase, the average of eigenstate overlaps $\overline{\mathcal{O}}$ exponentially decays with an increase in the system size, indicating the fragility of its eigenstates,and this can be considered as an eigenstate version of the butterfly effect of chaotic systems. For localized systems, $\overline{\mathcal{O}}$ is almost size independent, showing the strong robustness of the eigenstates and the inconsistency of the eigenstate thermalization hypothesis. In addition, we find that the response of eigenstates to the change in the boundary conditions in many-body localized systems is identified with the single-particle wave functions in Anderson localized systems. This indicates that the eigenstates of many-body localized systems, as many-body wave functions, may be independent of each other. We demonstrate that this is consistent with the existence of a large number of quasilocal integrals of motion in the many-body localized phase. Our results provide another method to study localized and delocalized systems from the perspective of eigenstates.