Hybrid-type nodal ring phonons and coexistence of higher-order quadratic nodal line phonons in an AgZr alloy

Nodal line states in electronic systems can be classified based on the order of dispersion around a band degeneracy (i.e., linear, quadratic, or cubic) and the various slopes of crossing bands (i.e., type I, type II, critical type, and hybrid type). The concept of topology has been extended to phonons, promoting the birth of topological phonons. A series of nodal line phonons have been predicted in solid-state materials both theoretically and experimentally. However, the proposed nodal line phonons have a type-I or type-II linear phonon band dispersion. This study proves, based on first principles, that the hybrid-type linear nodal ring phonons and the higher-order quadratic nodal line (QNL) phonons coexist in one realistic solid-state material---AgZr with P4/nmm structure. Moreover, an effective Hamiltonian characterizing the hybrid nodal ring (HNR) phonons and the QNL phonons was established to explore the relevant physics. The AgZr material can be viewed as an ideal material to study the entanglement between HNR and QNL phonons.