Robustness and Independence of the Eigenstates with respect to the Boundary Conditions across a Delocalization-Localization Phase Transition

TL;DR

This paper investigates how many-body eigenstates respond to boundary condition changes across a localization-delocalization transition, revealing their robustness or fragility and linking many-body localization to single-particle localization properties.

Contribution

It introduces eigenstate overlaps as a new measure to characterize eigenstate robustness and connects many-body localization features to single-particle wave functions and integrals of motion.

Findings

Eigenstate overlaps decay exponentially in the ergodic phase, indicating fragility.

Eigenstate overlaps are size-independent in localized systems, showing robustness.

Many-body localized eigenstates resemble single-particle localized wave functions.

Abstract

We focus on the many-body eigenstates across a localization-delocalization phase transition. To characterize the robustness of the eigenstates, we introduce the eigenstate overlaps $\mathcal{O}$ with respect to the different boundary conditions. In the ergodic phase, the average of eigenstate overlaps $\bar{\mathcal{O}}$ is exponential decay with the increase of the system size indicating the fragility of its eigenstates, and this can be considered as an eigenstate-version butterfly effect of the chaotic systems. For localized systems, $\bar{\mathcal{O}}$ is almost size-independent showing the strong robustness of the eigenstates and the broken of eigenstate thermalization hypothesis. In addition, we find that the response of eigenstates to the change of 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 the many-body localized systems, as the 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 a new method to study localized and delocalized systems from the perspective of eigenstates.