Ideal nodal net phonons in P n 3 ¯ m type Ag2O

The topological phonon state is a new field that has sparked much interest. Weyl phonons in FeSi, for example, have been theoretically proposed and experimentally identified. In this work, via the first-principle calculation, we prove the ideal nodal net phonons exist in a realistic material Ag2O with P n 3 ¯ m type structure. With the help of the Berry phase calculations, we find that the nodal net phonons in Ag2O are topologically non-trivial. The phononic surface states are visible, which benefits the experimental detections. The results in this work contribute to the material realization of topological nodal net phonons. The author hopes the experimental verification of the novel topological phonons can be performed in the following investigations.

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 Pn3m space group. Ag 1 ⁺ is bonded in a linear geometry to two equivalent O 2 ⁻ atoms. O 2 ⁻ is bonded to four equivalent Ag 1 ⁺ atoms to form corner-sharing OAg₄ tetrahedra. The lattice constants for the cubic Ag 2 O are optimized via firstprinciple 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 Ag 2 O is shown in Figure 1A, where the Ag atoms and O atoms occupy the 2a, and 4c Wyckoff positions, respectively.

Methods
The calculations for the realistic material Ag 2 O 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.

Weyl nodal net phonons
Based on the determined lattice constants, the phonon dispersion for a 2 × 2 × 2 supercell of Ag 2 O 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 Ag 2 O. 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 Ag 2 O, 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 Ag 2 O 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.

Phononic surface states
To examine the topological nature of the Weyl nodal net phonons in Ag 2 O, we calculate its Berry phase using the following formula: γ n C A n (k) · dl, where A n (k) −i n 〈u n (k)|∇ k |u n (k)〉 is the Berry connection. Calculated results show that the Weyl nodal net phonons host a π Berry phase, indicating topologically non-trivial behaviors. The nontrivial nodal net phonons will lead to phononic drumhead-like surface states. As shown in Figure 3D, we project the two Weyl points (belong to the Weyl net) intoR andX surface points to the (010) surface (see Figure 2B). The phonon LDOS projected on the (010) surface BZ alongR-X. Obviously, a clear drumhead-like surface state, connected by the projections of the Weyl points, is visible and marked by the black arrow. The bulk stats do not cover such a phononic surface state and would benefit the experimental detections shortly.

Summary and remarks
We show that nodal net phonons exist in Ag 2 O using firstprinciples calculations. Straight lines constrained in the highsymmetry line X-M at the BZ boundary represent the nodal net. Because there is no spin in phononic systems, the nodal net phonons in Ag 2 O 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: Frontiers in Physics frontiersin.org 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.

Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions
Calculating, writing, and researching are done by YL.

Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The reviewer HZ declared a shared affiliation with the author(s) add initials here unless all authors are concerned to the handling editor at the time of review.

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