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