Weyl-Dirac Nodal Lines

• Weyl-Dirac nodal lines are composite topological features combining twofold and fourfold band crossings enforced by specific crystalline symmetries.
• Model Hamiltonians illustrate how linear WNL terms and Dirac-like DNL contributions yield distinct Berry phase signatures, with π and 2π phases respectively.
• Material realizations in systems like KCuS and tailored metamaterials exhibit unique drumhead and torus surface states, underpinning innovative device applications.

Weyl-Dirac nodal lines are composite topological features in the band structures of crystalline solids and bosonic systems, characterized by the coexistence and interplay of Weyl nodal lines (WNLs)—lines of twofold-degenerate band crossings with quantized π Berry phase—and Dirac nodal lines (DNLs)—lines of fourfold-degenerate crossings with a 2π Berry phase, both typically enforced by space group symmetry. Their distinction and connectivity arise from crystalline symmetries such as nonsymmorphic operations, combined inversion and time-reversal, and mirror reflections. Systems exhibiting these nodal complexes present a setting for diverse surface states and unique bulk-boundary correspondence, especially in the context of topological phonons, fermions, and artificial metamaterials (Wang et al., 2021, Du et al., 2 Feb 2026, Liu et al., 2017).

1. Definitions and Symmetry Principles

A Weyl nodal line (WNL) consists of a continuous one-dimensional manifold in momentum space where two non-degenerate bands cross linearly, carrying a quantized Berry phase γ = π around a loop encircling the line. A Dirac nodal line (DNL) is a one-dimensional manifold of fourfold-degenerate crossings, where each point corresponds to the intersection of two Kramers-like doublets, protected by an antiunitary symmetry (such as $\mathcal{PT}$ or nonsymmorphic operations) resulting in a doubling of degeneracy. Around a loop enclosing a DNL, the Berry phase is γ = 2π (modulo 2π).

Weyl-Dirac nodal lines ("WDNLs", Editor's term) refer to configurations where DNLs and WNLs interconnect—typically, a straight fourfold DNL along a high-symmetry line is bridged or terminated by hourglass-like WNLs lying in high-symmetry planes. This complex structure is enforced by a specific set of crystalline symmetries only realized in a small subset of space groups, most notably Pnma (#62), as shown by group-theoretic classification (Du et al., 2 Feb 2026, Wang et al., 2021).

Symmetry Requirements

Nodal Line Type	Minimal Symmetry Protection	Typical Space Groups
Dirac (4-fold)	$\mathcal{PT}$, nonsymmorphic ops	Pnma, tetragonal SGs 127–138
Weyl (2-fold)	Mirror or glide only	Pnma, select orthorhombic SGs
Weyl-Dirac	Combination, interlocking IRRs	Pbcm, Pbcn, Pbca, Pnma, Pa3̄

Only five of the 230 space groups admit enforced WDNL networks in bosonic systems (phonons): Pbcm, Pbcn, Pbca, Pnma, and Pa3̄ (Du et al., 2 Feb 2026).

2. Model Hamiltonians and Band Topology

The low-energy effective Hamiltonians capture the symmetry and topological content of these nodal lines. WNLs are described by a generic two-band Dirac-like Hamiltonian,

$$H_W(\boldsymbol{q}) = v_x q_x \sigma_x + v_y q_y \sigma_y,$$

where $\sigma_i$ are Pauli matrices and $\boldsymbol{q}$ is the momentum measured from the nodal line. For DNLs, one employs a four-band Dirac-type form,

$$H_D(\boldsymbol{q}) = v_1 q_{\parallel} (\tau_0 \otimes \sigma_x) + v_2 q_z (\tau_z \otimes \sigma_y),$$

with $\sigma$ and $\tau$ labeling the Kramer-like subspaces.

In composite WDNL systems, compatibility relations among little-group irreducible representations force the coexistence and interconnection of these nodal lines—e.g., in Pnma, a four-dimensional representation on the S–R line enforces a DNL, while band crossings throughout the k_z = 0 plane generate an hourglass WNL intersecting the DNL at high-symmetry points (Du et al., 2 Feb 2026, Wang et al., 2021).

The topological invariant of a WNL is a π Berry phase for any small loop encircling the line. For a DNL—effectively two superimposed WNLs of opposite chirality—the Berry phase is 2π; these lines carry no net Chern number, but locally each pairwise crossing on the DNL can be decomposed into a doublet of π-charged WNLs (Du et al., 2 Feb 2026, Wang et al., 2021, Liu et al., 2017).

3. Geometric Structure and Material Realizations

The archetypal WDNL configuration consists of:

• Dirac nodal line: a straight, high-symmetry line (e.g., S–R in Pnma) with fourfold-degenerate crossings.
• Hourglass Weyl nodal line: a loop entirely in a high-symmetry plane (e.g., k_z = 0) exhibiting “hourglass” partner-switching band connectivity arising from the nonsymmorphic symmetry algebra—endpoints (necks of the hourglass) are pinned to high-symmetry points and are symmetry-immovable.
• Interconnection: the DNL and HWNL intersect at special points (e.g., the S-point), forming a composite nodal net or ladder in momentum space.

KCuS with Pnma symmetry is the first experimentally characterized phononic system to realize both HWNL and DNL, manifesting both in the isolated 5.0–5.2 THz phonon window (Wang et al., 2021). Other theoretically identified materials with WDNL phonons include NdRhO₃ and ZnSe₂O₅ (Du et al., 2 Feb 2026). In the electronic sector, TiB₂ (P6/mmm) features interlinked Dirac nodal rings, while systems such as Ba₅In₄Bi₅ show nodal chains with enforced crossings by nonsymmorphic symmetry (Liu et al., 2017, Hirschmann et al., 2021).

4. Surface-State Phenomenology

WDNLs give rise to distinct and surface-selective boundary states, depending on whether the surface projection intersects the WNL, the DNL, or both. The topological origin is the mismatch of Zak phase (surface-probed Berry phase) associated with the bulk lines:

• Drumhead (arc) surface states: flat or weakly dispersive bands filling the interior of the WNL projection on the surface Brillouin zone, occurring where the line normal to the surface carries π Zak phase; characteristic of WNL projections.
• Torus surface states (TSS): extended, dispersive surface bands that occupy the entire projected surface BZ or “wrap” around regions divided by the DNL; they occur when both surface momentum subdomains have nontrivial Zak phase, as with DNL surface projections (Du et al., 2 Feb 2026).

For example, in KCuS, only [100] and [001] surfaces (not [010]) harbor drumhead modes, reflecting the surface-specific Zak phase profile established by the bulk nodal structure (Wang et al., 2021). In lattice models and ARPES measurements, the drumhead modes appear as high-density, nearly flat patches bounded by the surface-projected nodal line (Liu et al., 2017, Hirschmann et al., 2021).

5. Topological Invariants and Bulk-Boundary Correspondence

The primary topological invariants are:

• Berry phase (γ): For WNL: γ = π, for DNL: γ = 2π (but trivial mod 2π; the local topology is always built from π-phase doublets).
• Surface Zak phase (ϕ_Zak): The in-plane Bloch momentum-resolved Zak phase distinguishes regions with surface polarization (and thus surface states), predicted by integrating the Berry connection along the surface-normal direction.
• Mirror/grading indices: Nontrivial winding in mirror or glide eigenvalues across the nodal line loci provides additional integer invariants, constraining the surface mode content (Wang et al., 2021, Du et al., 2 Feb 2026).

Bulk-boundary correspondence in WDNL phases is highly surface-selective and topologically robust, as the boundary states reflect the bulk Berry-phase structure and the detailed symmetry-resolved protection of each nodal component.

6. Experimental Realizations and Tunability

Several physical platforms and realizations have demonstrated or are predicted to exhibit WDNLs:

• Phononic crystals: KCuS, NdRhO₃, ZnSe₂O₅ feature coexisting HWNLs and DNLs in clean frequency windows, with surface-selective topological phonon states measurable via surface phonon spectral functions and surface phonon probes (Wang et al., 2021, Du et al., 2 Feb 2026).
• Electronic solids: Nodal networks are reported in AlB₂-type diborides (e.g., TiB₂), perovskites (Eu₃PbO), and tetragonal/intermetallic materials such as Ba₅In₄Bi₅, with ARPES and quantum oscillations mapping the surface “drumhead” and bulk-linked Fermi structures (Liu et al., 2017, Hirschmann et al., 2021, Hirschmann et al., 2022).
• Synthetic/metamaterial systems: Microwave photonic structures and acoustic metamaterials realize nodal chains and links, extending the framework to artificial platforms with high controllability (Yan et al., 2017, Park et al., 2022).
• Tunability: External strain, magnetic order, or structural distortion can move, create, or annihilate nodal lines and control the connectivity of WDNL networks, as seen in Weyl–Dirac crossover scenarios and field-tunable platforms like Eu₃PbO (Hirschmann et al., 2022, Du et al., 2 Feb 2026).

7. Broader Context and Outlook

The existence of Weyl-Dirac nodal lines establishes a generic paradigm for multi-topological-state band complexes beyond simple Dirac or Weyl semimetals. Their occurrence is predicated on stringent symmetry criteria, leading to rarity but also to exceptional phenomenology, including:

• The entanglement and transition between distinct topological “flavors” (π- and 2π-phase lines) within one system, enabling studies of bulk state interplay and hybridization.
• Surface states with termination-selective topological characteristics—potential for tailored surface response, high-density states, and device-relevant features such as selective surface localization of vibrational or electronic states.
• Opportunities for non-Abelian nodal line networks in three or more bands, with recent theoretical work addressing quaternion-valued topological charges and their possible physical consequences (Park et al., 2022).

Current challenges include establishing a rigorous bulk-surface correspondence for non-Abelian invariants, realizing fully clean nodal networks at electronic Fermi levels, and leveraging these nodal platforms for nonlinear, optoelectronic, and phononic device functionalities (Du et al., 2 Feb 2026, Wang et al., 2021, Park et al., 2022).

For detailed first-principles methodologies, symmetry analyses, and candidate material parameters, see (Wang et al., 2021, Du et al., 2 Feb 2026). For experimental ARPES evidence and mapping of nodal networks, see (Liu et al., 2017). For superlattice and synthetic implementations, photonic and phononic crystalline platforms provide robust, tunable WDNL analogues (Yan et al., 2017, Park et al., 2022).