Diagnosis of enforced phonon band crossings in the entire frequency window by compatibility relations

Topological band crossings (BCs) such as Dirac and Weyl points can provide intriguing physical properties in both fermionic systems and bosonic systems, like surface Fermi arcs, chiral anomaly, spin-momentum locking, and among others. Unlike BCs at high-symmetry points (HSPs) formed by a single degenerate irreducible representation (irrep), those in high-symmetry lines (HSLs) are degeneracies formed by two different irreps. For fermionic systems, such BCs near the Fermi level can be effectively diagnosed using symmetry data at HSPs, namely, the number of times each irrep of occupied bands occurs at HSPs. For phonon systems, phonon BCs in the entire frequency window are meaningful. Here, we propose a new diagnostic scheme for detecting phonon BCs by compatibility relations (CRs), which would otherwise be invisible to traditional methods as applied in electronic bands. Based on 230 space groups, we apply our diagnostic scheme to the Phonon Database at Kyoto University (consisting of 10,034 materials) and identify 2,815,357 emergent particles (EMPs) in HSLs. These EMPs include C-1 WP, C-2 DP, C-2 WP, C-3 WP, DP, P-WNL, P-WNLs, QDP, QTP, and TP, providing a platform to assist experimentalists in exploring the practical application value of phonon EMPs. Our diagnostic scheme can also be applied to other bosonic systems like photon and magnon systems and extended to other symmetries, such as magnetic space groups and spin space groups.