Unconventional phonon spectra and obstructed edge phonon modes

Based on the elementary band representations (EBR), many topologically trivial materials are classified as unconventional ones (obstructed atomic limit), where the EBR decomposition for a set of electronic states is not consistent with atomic valence-electron band representations. In the work, we identify that the unconventional nature can also exist in phonon spectra, where the EBR decomposition for a set of well-separated phonon modes is not consistent with atomic vibration band representations (ABR). The unconventionality has two types: type I is on an empty site; and type II is on an atom site with non-atomic vibration orbitals. The unconventionality is described by the nonzero real-space invariant at the site. Our detailed calculations show that the black phosphorus (BP) has the type I unconventional phonon spectrum, while 1H-MoSe2 has the type II one, although their electronic structures are also unconventional. Accordingly, the obstructed phonon modes are obtained for two types of unconventional phonon spectra.