Characterizing many-body localization via state sensitivity to boundary conditions

We introduce characterizations for many-body phase transitions between delocalized and localized phases based on the system's sensitivity to boundary conditions. In particular, we change conditions from periodic antiperiodic calculate a shift in energy shifts single-particle density-matrix eigenvalues corresponding window. employ typical model studying MBL, one-dimensional disordered system of fermions with nearest-neighbor repulsive interaction where disorder is introduced as randomness on-site energies. By calculating numerically density matrix, observe that regime, both are vanishing; while extended order level spacing. also applied these transition case having next-nearest-neighbor interactions addition studied its effect transition.