A tale of two localizations: coexistence of flat bands and Anderson localization in a photonics-inspired amorphous system

Emerging experimental platforms use amorphousness, a constrained form of disorder, to tailor meta-material properties. We study localization under this type of disorder in a family of 2D models generalizing recent experiments on photonic systems. Models in this family reside on amorphous analogs of kagom\'{e} lattices with fixed coordination number, vary by a tunable synthetic field, and remarkabaly, permit exact results. We observe two kinds of localization that emerge in these models: Anderson localization by amorphous disorder, and the existence of compact, macroscopically degenerate localized states as in many crystalline flat bands. The flat-band-like degeneracy innate to kagom\'{e} lattices survives under amorphousness without on-site disorder. This phenomenon arises from the cooperation between the structure of the compact localized states and the geometry of the amorphous graph. More surprisingly, for particular values of the field, such states emerge in the amorphous system that were not present on the kagom\'{e} lattice in the same field. Outside the flat band, constrained amorphous graph geometry necessitates the existence of a fully delocalized state, near which we observe evidence of a localization-delocalization transition. Our platform serves as a demonstration of how the qualitative behavior of a disordered system can be tuned at fixed graph topology and lead to localization phenomena unique to amorphous systems that are not observed in their generically disordered counterparts.