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Abstract The localized nature of a flat band is understood by the existence of a compact localized eigenstate. However, the localization properties of a partially flat band, ubiquitous in surface modes of topological semimetals, have been unknown. We show that the partially flat band is characterized by a non-normalizable quasi-compact localized state (Q-CLS), which is compactly localized along several directions but extended in at least one direction. The partially flat band develops at momenta where normalizable Bloch wave functions can be obtained from a linear combination of the non-normalizable Q-CLSs. Outside this momentum region, a ghost flat band, unseen from the band structure, is introduced based on a counting argument. Then, we demonstrate that the Wannier function corresponding to the partially flat band exhibits an algebraic decay behavior. Namely, one can have the Wannier obstruction in a band with a vanishing Chern number if it is partially flat. Finally, we develop the construction scheme of a tight-binding model for a topological semimetal by designing a Q-CLS.
Park et al. (Tue,) studied this question.