We show that the rainbow state, which has volume law entanglement entropy for most choices of bipartitions, can be embedded in a many-body localized spectrum. For a broad range of disorder strengths in the resulting model, we numerically find a narrow window of highly entangled states in the spectrum, embedded in a sea of area law entangled states. The construction hence embeds mobility edges in many-body localized systems. This can be thought of as the complement to manybody scars, an 'inverted quantum many-body scar', providing a further type of setting where the eigenstate thermalization hypothesis is violated.

Closed 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 mean entropy as a function of disorder strength for eigenstates in the middle of the spectrum and by observing that the adjacent gap ratio undergoes a transition from the value for Wigner-Dyson statistics to the value for Poisson statistics as the disorder strength is increased. When the Hamiltonian is perturbed in such a way that the special state is no longer an eigenstate, the weak violation of MBL disappears, which suggests that the partial solvability of the model together with the particular form of the state are the source of the violation.