  Blending ordering within an uncorrelated disorder potential in families of 3D
Lieb lattices preserves the macroscopic degeneracy of compact localized states
and yields unconventional combinations of localized and delocalized phases --
as shown in Phys.Rev.B 106, 214204 (2022). We proceed to reintroduce
translation invariance in the system by further ordering the disorder, and
discuss the spectral structure and eigenstates features of the resulting
perturbed lattices. We restore order in steps by first (i) rendering the
disorder binary -- i.e. yielding a randomized checkerboard potential, then (ii)
reordering the randomized checkerboard into an ordered one, and at last (iii)
realigning all the checkerboard values yielding a constant potential shift, but
only on a sub-lattice. Along this path, we test the influence of additional
random impurities on the order restoration. We find that in each of these
steps, sub-families of states are projected upon the location of the degenerate
compact states, while the complementary ones are localized in the perturbed
sites with energy determined by the strength of checkerboard. This strategy,
herewith implemented in the 3D Lieb, highlights order restoration as
experimental pathway to engineer spectral and states features in disordered
lattice structures in the pursuit of quantum storage and memory applications.
