Sixfold degenerate nodal-point phonons: Symmetry analysis and materials realization

Abstract

Multifold degenerate fermions in electronic structures of topological materials give rise to many interesting physics properties. Among them, three-, six-, and eightfold degenerate fermions in condensed matter systems are considered important because these fermions are not allowed in high-energy physics due to restriction posed by Poinc\'are symmetry. Phonons are the basic emergent boson of the crystalline lattice. Moreover, topological phonons also exist in crystalline solids, like fermionic electrons, due to the crystal symmetry constraints. Two-, three-, and fourfold degenerate phonons were predicted previously. This study proposes degenerate phonons with the maximum fold, i.e., sixfold; we find them in five space groups (with numbers 218, 220, 222, 223, and 230) through a detailed symmetry analysis. We also propose a series of realistic materials with above-mentioned space group numbers that host sixfold degenerate nodal-point phonons at the high-symmetry points $H$ (or $R$) point using first-principles calculations. Finally, as examples, we investigate the surface phonon spectrum of ${\mathrm{C}}_{3}{\mathrm{N}}_{4}$, ${\mathrm{Sc}}_{4}{\mathrm{C}}_{3}$, ${\mathrm{Y}}_{4}{\mathrm{Sb}}_{3}$, and ${\mathrm{K}}_{8}{\mathrm{Si}}_{46}$ materials and find two obvious projected phonon surface states on (110) or (100) surfaces.