  Isolated quantum systems typically follow the eigenstate thermalization
hypothesis, but there are exceptions, such as many-body localized (MBL) systems
and quantum many-body scars. Here, we present the study of a weak violation of
MBL due to a special state embedded in a spectrum of MBL states. The special
state is not MBL since it displays logarithmic scaling of the entanglement
entropy and of the bipartite fluctuations of particle number with subsystem
size. In contrast, the bulk of the spectrum becomes MBL as disorder is
introduced. We establish this by studying the entropy as a function of disorder
strength and by observing that the level spacing statistics undergoes a
transition from Wigner-Dyson to Poisson statistics as the disorder strength is
increased.
