Unconventional delocalization in a family of three-dimensional Lieb lattices

Uncorrelated disorder in generalized three-dimensional 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, and 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.