Unconventional delocalization in a family of 3D Lieb lattices

Uncorrelated disorder in generalized 3D Lieb models gives rise to the existence of bounded mobility edges, destroys the macroscopic degeneracy of the flat bands and breaks their compactly-localized states. We now introduce a mix of order and disorder such that this degeneracy remains and the compactly-localized states are preserved. We obtain the energy-disorder phase diagrams and identify mobility edges. Intriguingly, for large disorder the survival of the compactly-localized states induces the existence of delocalized eigenstates close to the original flat band energies -- yielding seemingly divergent mobility edges. For small disorder, however, a change from extended to localized behavior can be found upon decreasing disorder -- leading to an unconventional ``inverse Anderson" behavior. We show that transfer matrix methods, computing the localization lengths, as well as sparse-matrix diagonalization, using spectral gap-ratio energy-level statistics, are in excellent quantitative agreement. The preservation of the compactly-localized states even in the presence of this disorder might be useful for envisaged storage applications.