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

TL;DR

This work addresses the coexistence of Weyl and Dirac nodal lines in phonon bands by developing a group-theoretical framework that rapidly identifies Weyl-Dirac NL phonons, revealing that only five space groups can host such states. A two-step approach first locates Dirac NLs along high-symmetry lines via $4$-dimensional IRRs, then uses compatibility relations to uncover Weyl NLs on high-symmetry planes and their connectivity, yielding a composite nodal network with distinct Berry phases: $\pi$ for Weyl NLs and $2\pi$ for Dirac NLs. The authors validate the framework through first-principles phonon calculations in NdRhO$_{3}$ (SG $62$) and ZnSe$_{2}$O$_{5}$ (SG $60$), predicting termination-selective surface states—drumhead states associated with Weyl NLs and torus-like surface states linked to Dirac NLs—on specific crystal facets. The findings enable type-selective surface state engineering in phononic systems and offer a practical route to explore the interplay of multiple topological states in bosonic lattices, with NdRhO$_{3}$ and ZnSe$_{2}$O$_{5}$ as concrete experimental candidates.

Abstract

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.

Figures (6)

• Figure 1: Schematic figures showing three types of nodal lines (NLs). The orange region denotes high-symmetry plane. (a) Weyl nodal line (red circle) with a Berry phase of $\pi$. (b) Straight Dirac nodal line (blue solid line) along a high-symmetry line with a Berry phase of $2\pi$. (c) A possible Weyl-Dirac NL structure.
• Figure 2: Schematic diagram of the WNL formation mechanism. (a) The irreducible BZ for the orthorhombic space group. C, D, E, and P (V and W) denote generic points along high-symmetry lines (on high-symmetry planes). (b) Schematic energy bands along the C-V-D path in the $k_{z}=0$ plane. (c) Schematic energy bands along the E-W-P path in the $k_{z}=\pi$ plane. The IRRs $V_{1}$ ($W_{1}$) and $V_{2}$ ($W_{2}$) are indicated by solid lines of different colors (red and blue).
• Figure 3: (a) Crystal structure of NdRhO$_3$. (b) Bulk and surface BZs. (c) Schematic showing Dirac-Weyl NL. The Dirac (Weyl) NL is indicated by blue (red) solid line. (d) Phonon spectrum of NdRhO$_3$ along high-symmetry paths. (e) Enlarged phonon dispersions of blue regions in (d). The red dots mark the band crossings forming the WNLs.
• Figure 4: The surface properties of NdRhO$_3$. (a)-(c) Schematics of the surface states for the (001), (010), and (100) surfaces. Red and blue lines indicate projections of bulk WNLs and DNLs onto the surface BZ, respectively. The values 0 and $\pi$ represent the Zak phases for lines normal to each surface. (d)-(f) Calculated surface spectra on the (001), (010), and (100) surfaces. (d) shows the drumhead surface states, while (f) shows the TSS.
• Figure 5: (a) Crystal structure of ZnSe$_2$O$_5$ material. (b) Bulk and surface BZs. (c) Schematic illustration of the Dirac (Weyl) nodal lines, indicated by blue (red) solid lines. (d) Phonon dispersion of ZnSe$_2$O$_5$ along high-symmetry paths. (e) Enlarged view of the phonon dispersion in the blue regions of (d). Red dots mark the band crossings forming the WNLs.