Title: Quantum engineering for compactly localized states in disordered Lieb lattices
Abstract: 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 models, 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.