Revisiting Flat bands and localization

Flat bands imply lack of itinerancy due to some constraints that, in principle, results in anomalous behaviors with randomness. By a molecular orbital (MO) representation of the flat band systems, random MO models are introduced where the degeneracy due to the flat bands is preserved even with randomness. The zero modes of the chiral symmetric system with sublattice imbalance belong to the class. After explaining the generic flat band construction by MOs, several examples are discussed with numerical demonstration as sawtooth lattice in one dimension and hyper-Pyrochlore lattice in any $d$-dimensions that extends the Kagome ($d=2$) and Pyrochlore ($d=3$) lattices to general dimensions.