Topological semimetals [1–10] with symmetry-protected bands crossing around the Fermi level have inspired enormous interest in condensed matter physics. As a typical family of topological semimetals, node-line semimetals [11–20] have high band degeneracy along a certain line in the Brillouin zone (BZ), and the resultant drumhead surface states at the boundary. Suppose more than one nodal line/ring appears in momentum space. In that case, these nodal lines/rings may form complex topological nodal structures in the three-dimensional (3D) Brillouin zone (BZ), such as nodal-chain [21–28], nodal-box [29], nodal-link [30–35], and nodal-net [36, 37] structures.

Topological state research has recently been extended to bosonic systems, such as photons in photonic crystals, phonon systems in 3D solids, and classical elastic waves in macroscopic artificial phononic crystals. In realistic materials, topological phonons [38–45] could play an essential role in thermal transports, electron-phonon coupling, and other phonon-related processes. Thus far, researchers have predicted some realistic materials [46–65] to host nodal net phonons. For example, using first-principles calculations, [66] discovered a nodal net state in the semiconductor copper chloride (CuCl). CuCl has a cubic crystal structure with the space group Pa3 (No. 205). In momentum space, the nodal net has a hexahedral shape and is made up of interconnected quadruple degenerate straight nodal lines. Moreover, with the help of first-principle calculations and symmetry analysis, [67] proposed the coexistence of the three-nodal surface and nodal net phonons in space groups with numbers 61 and 205. Some realistic materials, such as ZnSb with SG No. 61 and RuS₂, P₂Pt, and OsS₂ with SG No. 205, hosting three-nodal surface and nodal net phonons have also been identified by [67].

In this paper, based on first-principles calculations, we contribute to one more realistic material with ideal nodal net phonons. Ag₂O is Cuprite structured and crystallizes in the cubic P n 3 ¯ m space group. Ag¹⁺ is bonded in a linear geometry to two equivalent O²⁻ atoms. O²⁻ is bonded to four equivalent Ag¹⁺ atoms to form corner-sharing OAg₄ tetrahedra. The lattice constants for the cubic Ag₂O are optimized via first-principle calculations. The obtained results from the calculations for the lattice constants are a = b = c = 4.81 Å, which are in good agreement with the experimental data [68], i.e., a = b = c = 4.73 Å. The crystal structure of the relaxed Ag₂O is shown in Figure 1A, where the Ag atoms and O atoms occupy the 2a, and 4c Wyckoff positions, respectively.

The calculations for the realistic material Ag₂O were performed using the Vienna ab initio Simulation Package [69] and the framework of density functional theory. The calculation’s energy and force convergence criteria were set to 10⁻⁶ eV and −0.01 eV/A, respectively. The plane-wave expansion was truncated at 500 eV, and the entire BZ was sampled by a 7 × 7 × 7 Monkhorst-Pack grid. We used the PHONOPY code to generate the symmetry information and construct the constant force matrices for phonon spectra calculations. To calculate the phonon surface states, we used the WannierTools package [70] in conjunction with the iterative Green function method to construct the tight-binding model Hamiltonian.

Based on the determined lattice constants, the phonon dispersion for a 2 × 2 × 2 supercell of Ag₂O is shown in Figure 2A. Note that the phonon-related properties in this work are calculated based on the density functional perturbation theory (DFPT). According to Figure 2A, there is no imaginary frequency in the phonon spectrum, indicating the dynamical stability of cubic Ag₂O. Around the frequency of five THz, two phonon bands along X-R-Γ-M and Γ-X merged into one twofold degenerate phonon bands along the X-M path. We want to point out that a similar case can also be found around the frequency of 14 THz. Here, we only focus on the twofold degenerate phonon bands along the X-M path. The three-dimensional plot of the twofold degenerate phonon bands along the X-M is shown in Figure 2B, and one can see a cross-shaped nodal structure appears (see the dotted lines) in Figure 2B.

To better view the cross-shaped nodal structure in the phonon dispersion of Ag₂O, we plot the phonon band structure of the crossing branches in ky = π plane (See Figure 3A). The cross-shaped nodal structure in the k_y = π plane is formed from crossing two straight Weyl nodal lines. We select some symmetry points along the X-M path and show the phonon dispersions for Ag₂O along the a-b-a’, c-d-c’, e-f-e’, and g-h-g’ in Figure 3C. All the crossing points are doubly degenerate points with a linear phonon band dispersion (see Figure 3C). Note that the cross-shaped nodal structures can also exist in k_y = -π, k_x = ± π, k_z = ± π plane. That is, these straight nodal lines are perpendicular to each other in different planes, forming a Weyl nodal net in the 3D BZ, as illustrated in Figure 3B.

We show that nodal net phonons exist in Ag₂O using first-principles calculations. Straight lines constrained in the high-symmetry line X-M at the BZ boundary represent the nodal net. Because there is no spin in phononic systems, the nodal net phonons in Ag₂O are resistant to time-reversal symmetry breaking. Before closing the paper, we would like to point out that our results can also guide the investigations of the Weyl nodal net in spinless electronic systems (such as topological semimetals without the consideration of spin-orbital coupling). We present a spinless lattice model to demonstrate the existence of Weyl nodal net states in spinless materials with SG 224. A unit cell with one site (0,0,0) was considered in this spinless lattice model, and s orbital was placed on this site. The four-band tight-binding (TB) Hamiltonian is shown as follows: H = e A B C A e D E B D e F C E F e , where A = 4 s ⁡ Cos ⁡ k x Cos ⁡ k y + k z / 2 , B = 4 s ⁡ cos ⁡ ⁡ k y Cos ⁡ k x + k z / 2 , C = 4 s ⁡ Cos ⁡ k x + k y / 2 Cos ⁡ k z , D = 4 s ⁡ Cos ⁡ k x − k y / 2 Cos ⁡ k z , E = 4 s ⁡ Cos ⁡ k y Cos ⁡ k x − k z / 2 , F = 4 s C o s ⁡ k x Cos ⁡ k y − k z / 2 . When we set e = 0.4 and s = 0.1, the band structure of this spinless lattice model is shown in Figure 4. From Figure 4, one finds that the appearance of the Weyl nodal line along the X-M path further reflects the occurrence of the Weyl nodal net in 3D BZ.