Weyl-Dirac nodal line phonons with type-selective surface states

The band complex formed by multiple topological states has attracted extensive attention for the emergent properties produced by the interplay among the constituent states. Here, based on group theory analysis, we present a scheme for rapidly identifying the Weyl-Dirac nodal lines (a complex of Weyl and Dirac nodal lines) in bosonic systems. We find only 5 of the 230 space groups host Weyl-Dirac nodal line phonons. Notably, the Dirac nodal line resides along the high-symmetry line, whereas the Weyl nodal line is distributed on the high-symmetry plane and is interconnected with the Dirac nodal line, jointly forming a composite nodal network structure. Unlike traditional nodal nets, this nodal network exhibits markedly distinct surface states on different surfaces, which can be attributed to the fundamental differences in the topological properties between the Weyl and Dirac nodal lines. This unique property thus allows the material to present distinct surface states in a termination-selective manner. Furthermore, by first-principles calculations, we identify the materials NdRhO${3}$ and ZnSe${2}$O$_{5}$ as candidate examples to elaborate the Weyl-Dirac nodal line and their related topological features. Our work provides an insight for exploring and leveraging topological properties in systems with coexisting multiple topological states.

💡 Research Summary

The authors investigate a new class of topological phonon excitations they term “Weyl‑Dirac nodal line (WDL) phonons,” which consist of coexisting Weyl nodal lines (WNLs) and Dirac nodal lines (DNLs) within a single bosonic band structure. Using a systematic group‑theoretical analysis of all 230 space groups, they first identify DNLs as four‑dimensional irreducible representations (IRRs) that appear along high‑symmetry lines (HSLs). This screening yields five candidate space groups (SG 57, 60, 61, 62, 205) that can host DNLs. The second step is to determine under what symmetry conditions a WNL can be forced to appear on a high‑symmetry plane (HSPL). The key requirement is the presence of glide‑mirror symmetries that protect four‑fold degeneracies on those planes. By examining SG 62 (Pnma) in detail, they show that the DNL lies on the SR line, while the glide‑mirror fMz guarantees a band crossing on any path that connects two HSLs adjacent to the DNL within the kz = 0 plane. Compatibility relations between the IRRs of the little groups at the HSLs and the HSPL enforce a mandatory crossing, which extends into a continuous WNL across the entire plane. The same reasoning applies to the other four space groups, and the authors compile a table listing the DNL locations, the WNL planes, the relevant compatibility relations, and the intersection points where the two nodal lines meet.

A crucial physical distinction between the two types of nodal lines lies in their Berry phases: a loop encircling a WNL carries a π phase, whereas a loop around a DNL carries 2π. Consequently, the surface manifestations differ: WNLs generate drumhead surface states, while DNLs give rise to torus‑shaped surface states. Because the projections of the WNL and DNL onto a given crystal facet are generally distinct, the surface spectrum becomes facet‑dependent. The authors term this “termination‑selective surface states,” meaning that by choosing a particular surface termination (e.g., (001), (010), or (100)), one can isolate either the drumhead or torus surface states.

To demonstrate the theory, the authors perform first‑principles density‑functional theory (DFT) and density‑functional perturbation theory (DFPT) calculations on two realistic compounds: NdRhO₃ (SG 62) and ZnSe₂O₅ (SG 205). For NdRhO₃, phonon calculations reveal a four‑fold degenerate DNL along the SR line (phonon branches 33–36) and a pair of crossing bands (34 and 35) along Γ–Y that belong to a WNL lying in the kz = 0 plane. The DNL and WNL intersect at the S point, forming the predicted composite nodal network. Surface Green’s‑function calculations based on Wannier‑derived tight‑binding models show that the (001) surface hosts drumhead states (projection of the WNL), whereas the (010) and (100) surfaces display torus‑type surface states (projection of the DNL). Similar features are identified in ZnSe₂O₅, confirming that the WDL phenomenon is not limited to a single material class.

Overall, the paper delivers three major contributions: (1) a rapid, group‑theory‑based protocol for identifying Weyl‑Dirac nodal line phonons across all space groups; (2) a clear exposition of how differing Berry phases lead to facet‑dependent surface signatures, enabling termination‑selective probing of topological phonons; and (3) concrete material realizations (NdRhO₃, ZnSe₂O₅) that are experimentally accessible. The work opens avenues for exploring multi‑topology bosonic systems, investigating temperature or pressure‑driven nodal‑line evolution, and coupling topological phonons to electrons or photons for hybrid quantum functionalities.