Symmetry-Enforced Weyl Phonons

In spinful electronic systems, time-reversal symmetry makes that all Kramers pairs at the time-reversal-invariant momenta are Weyl nodes in chiral crystals, which lack improper rotation symmetries, such as inversion or mirror. Similarly, such symmetry-enforced Weyls can also emerge in spinless systems ($e.g.$ phononic and optical systems) due to nonsymmorphic symmetries. In this work, we demonstrate that, for some nonsymmorphic chiral space groups, a certain high-symmetry $k$ point can only host Weyl points in the phononic systems, dubbed symmetry-enfored Weyl phonons (SEWPs). The SEWPs, enumerated in Table I, are located on the boundary of the 3D Brillouin zone and protected by nonsymmorphic crystal symmetries. By using first-principles calculations and symmetry analyses, we propose that, as an example of SEWPs, the two-fold degeneracies at P are monopole Weyl nodes in the crystal of K$_2$Sn$_2$O$_3$ in SG 199. The two nonequivalent P points are related by time-reversal symmetry and host the same chirality. In particular, at $\sim 20$ THz, the two Weyl phonons of the same chirality are compensated with a three-fold spin-1 Weyl phonon at the H point. The significant separation between P and H points guarantees the phonon surface arcs very long and clearly visible. This work not only presents a novel strategy to search for monopole Weyls in the spinless systems, but also offers some promising candidates for studying monopole Weyl and spin-1 Weyl phonons in realistic materials.

Topological phonons [1][2][3][4][5][6][7][8], referring to the quantized excited vibrational states of interacting atoms, have been most recently attracted attentions in condensed matter physics because of their unique physical nature [9][10][11][12][13][14]. In similarity to various fermions of electrons, topological phonons such as (spin-1/2 or monopole) Weyl, Dirac, spin-1 Weyl and charge-2 Dirac phonons have been predicted or observed in three-dimensional (3D) momentum space of solid crystals with topological vibrational states [15][16][17][18][19][20][21][22][23], strengthening largely our understanding of elementary particles in the universe. For instance, Zhang et al. predicted that both spin-1 Weyl phonons and charge-2 Dirac phonons exist in the CoSi system [23]. Spin-1 Weyl phonons are constructed by three bands with Chern numbers of 0 and ±2, while charge-2 Dirac phonons by four bands with two +1 and two −1. The coexistence of above two different classes of topological phononic quasiparticles exhibits exotic topological nontrivial features, such as noncontractable surface arcs and double-helicoid surface states [24]. Moreover, phonons can be excited to all energy space to generate unusual transport behaviors, since they are not limited by Pauli exclusion principle and Fermi surface in materials. These particular properties support their potential device applications, which drives researchers to explore new topological phononic states and ideal material candidates.
According to Nielsen-Ninomiya no-go theorem [25]. Weyl points (WPs) always appear in pairs at generic points, characterized by positive and negative features, acting as sources or sinks of Berry curvature distribu- The schematic symmetry-enforced spin-1/2 (or monopole) Weyl points (WPs). Even though a WP is stable in the 3D Brillouin zone, an addition symmetry can pin the WP at a certain momentum. a, In a spinful system of a chiral crystal, all the Kramers degeneracies at the timereversal invariant momenta (TRIM) are WPs. In a spinless system of a chiral crystal, the two-fold degeneracies can also be enforced to form WPs, by some non-symmorphic crystal (unitary) symmetries in b, or by the combined (anti-unitary) symmetry of TR symmetry and a four-fold screw symmetry-41 in c.
tions in the 3D Brillouin zone (BZ) [26]. They are usually not easy to be predicted due to the lack of symmetry protections, and the Weyl predictions usually require comprehensive numerical calculations in the BZ [27]. However, in the spinful electronic systems, Kramers pairs are enforced to be WPs by time-reversal (TR) symmetry at the time-reversal-invariant momenta in chiral crystals (Fig. 1a), where there is no improper rotation symmetry [16,17]. By checking carefully the symme-tries of 230 space groups (SGs), we have demonstrated that such symmetry-enforced Weyls can emerge in spinless systems, such as phononic systems (mainly focused on in the work) and optical systems, due to nonsymmorphic symmetries (Fig. 1b, c). These Weyls are all pinned at the high-symmetry points on the boundary of the 3D BZ. Do not like the TR-enforced Weyls in spinful electronic systems [16], where the Weyls are usually buried in bulk states due to the weak strength of spinorbital coupling, the nonsymmorphic-crystal-symmetryenforced Weyl phonons can be well isolated at some frequencies. Consequently, the associated Fermi arcs are long and robust, which can be easily probed in future experiments. In this work, by doing symmetry analyses in 230 SGs, we have demonstrated that, for some chiral SGs, a certain high-symmetry k-point can only host WPs in the phononic systems, dubbed symmetryenforced Weyl phonons (SEWPs). The SEWPs are located on the boundary of the 3D BZ and protected by nonsymmorphic crystal symmetries. We enumerate all the SEWPs at the high-symmetry points of the SGs in Table I. In this Table, all the phonon bands are doubly degenerate at those high-symmetry points and each twofold degeneracy represents a WP. Thus, one can easily predict Weyl phonons in such systems as long as the materials are of the SGs in Table I, which significantly lowers the difficulty to predict the Weyls in phononic systems. By employing the first-principles calculations, we predicted that, as an example of the SEWPs, two-fold degeneracies at the high-symmetry point P are WPs in the crystal of K 2 Sn 2 O 3 in SG 199. First, there are two nonequivalent P points in the first BZ, which are related by time-reversal symmetry. Therefore, the Weyls at two P points host the same chirality. Second, at ∼ 20 THz, the two Weyl phonons between 39 th and 40 th bands are compensated with a three-fold spin-1 Weyl phonon at the H point. Third, the spin-1 Weyl phonon here has been found to locate on the boundary of the BZ, which is immune to the LOTO (longitudinal and transverse optical phonon splitting) modification in the phonon spectrum. Lastly, the WPs at a certain energy level host the same chiral charge, giving rise to nontrivial Fermi surfaces associated with nonzero Chern numbers. It could generate the quantized circular photogalvanic effect [28][29][30][31]. In addition, the very long phonon surface arcs confirm further the robustness of the Weyl phonons. More examples of SEWPs can be found in Supplementary section C. These new findings not only provide a new strategy to search for monopole WPs in spinless systems, but also predict some promising candidates for studying topological quasiparticles in experiments.

Searching for SEWPs by symmetry analysis
The guiding principle of the search of SEWPs in the 230 SGs is: i) to find the high-symmetry k point of the space group, where the dimensions of all irreducible represen- I: The list of the SEWPs. The first and the second columns indicate the SG number and the corresponding highsymmetry point, respectively. The third and fourth columns show the abstract group (AG), to which the little group of the k-point is isomorphic, and the corresponding irreps (separated by semicolons). The k-point (uvw) ≡ ug1 + vg2 + wg3 and the translation {E|abc} ≡ T (at1 +bt2 +ct3) are given in units of the primitive reciprocal lattice vectors (i.e., g1, g2, g3) and primitive lattice vectors (i.e., t1, t2, t3). See more details for the AGs, their character tables and the definition of these lattice vectors in ref. [32]. tations (irreps) of the little group are two; ii) then to exclude the SGs that contain improper rotational symmetries, to make sure that there is no double degeneracy along any high-symmetry line/plane crossing the WPs.
Since we are interested in spinless systems, we consider only the single-valued representations; TR symmetry is an antiunitary that squares to 1. Table I summarizes the results of our search. Those high-symmetry WPs are on the boundaries of the 3D BZ. All of them are chiral SGs with non-symmorphic symmetries, and all representations are projective; these are in fact necessary ingredients for the (spin-1/2) Weyl excitations in the (spinless) phononic systems. We find that most of two-fold degeneracies are protected by the anti-commutation relation of two unitary operators {A, B} = 0, given in Table I )] = −1, which enforces a Kramers-like degeneracy as discussed in ref. [33].
Then, we take SG 199 as an example to illustrate the anti-commutation relation in the main text (see more derivations for all other SGs of Table I in Supplementary section C). SG 199 hosts only Weyl phonons at the P point (the high-symmetry points are defined in ref. [32]), even though it hosts three different irreps of the AG G 3 48 in Table I. This SG has a body-centered cubic Bravais lattice. The operators C 2x and C 2z acting on the primitive lattice vectors (t 1 , t 2 , t 3 ) are presented [32] as below: Thus, At the P point ( 1 4 , 1 4 , 1 4 ), the pure translation operator {E|3, 2, 1} is expressed as exp[2iπ(3 + 2 + 1)/4] = −1. Therefore, we get {A, B} = 0, which yields that all the phonon bands to be at least two-fold degenerate at the P point. Note that no higher n-fold (n > 2) degenerate irreps are found at P. In addition, we have checked that there is no symmetry-protected degeneracy on the highsymmetry planes/lines crossing the P point.
Effective k · p models Let's consider a two-band model at the P point in SG 199 first. We have A 2 = {E|000} = 1, B 2 = {E|220} = 1 with E identity operator and {A, B} = 0. With the matrix representations of A = σ x and B = σ z , the k · pinvariant Hamiltonian is derived as (to the first order), with σ x,y,z Pauli matrices, k x,y,z momentum offset from the P point, and v 1,2,3 real coefficients. Obviously, it's a Weyl Hamiltonian. Other SGs in Table I with antiunitary commutation relations share the similar results (see more in Supplementary section C). We notice that the P point is a TR non-invariant point at a corner of the BZ (that is, P = −P). Those systems host another Weyl phonons of the same chirality at −P due to TR symmetry, indicating that there must be some other nontrivial excitations in the system, to compensate the nontrivial Barry curvature arising from the two Weyls of the same chirality. For example, in the following example of K 2 Sn 2 O 3 in SG 199, a spin-1 Weyl phonon (C = 2) is found at the H point, to be consistent with two Weyl phonons (C = −1) at the P points. Then we consider the P point in SG 80, where we can not find two anti-commutation symmetry operators. We consider two symmetry operators at P point: D= {C 2z |100} and T · 4 1 , where 4 1 is a nonsymmorphic four-fold rotational symmetry, followed by a fractional lattice translation (T( c/4), where c is a lattice constant in the z direction). It is worth noting that (4 1 ) 2 =D and 4 1 is not a symmetry operator that keeps P invariant (See more details in Subsection 2 of Supplementary section C). We can express the two operators as and T · 4 1 = 1 √ 2 (σ x + σ y )K, which meet the conditions: (T ·4 1 ) 2 = D and (T ·4 1 ) 4 = −1. Thus, the k·p invariant Hamiltonian is derived as (to the first order), It's also a Weyl Hamiltonian with isotropy in the k x − k y plane.

SEWPs in realistic materials
The phonon dispersion of any material in the SG included in Table I has to contain WPs at those high-symmetry points. To confirm our theoretical results, we have systematically performed the ab-initio phonon calculations on some materials for each SEWP SG. As an example, we focus on the results and discussions on K 2 Sn 2 O 3 of SG 199 in the main text, and put other results in Supplementary section C. The crystallographic data of K 2 Sn 2 O 3 are obtained from ref. [34], and the primitive cell is illustrated in Fig. 2a, where the purple (black and red) atoms stand for K (Sn and O) atoms. The material example belongs to the body-centered cubic structures with SG group I2 1 3, which are in good agreements with previous experimental results [35]. Each primitive cell contains 14 atoms with four K, four Sn and six O atoms. The corresponding BZ is shown in Fig. 2b, where the blue and red squares are the (001) and (110) surface BZs, respectively.
The calculated phonon dispersion of K 2 Sn 2 O 3 is shown in Fig. 2c. It's clearly seen that there are some band crossings (degeneracies) at some high-symmetry points, especially at the high frequencies. First, we do find that all the degenerate phonon bands at the P point (at different frequencies) are WPs. The corresponding results of the Chern number calculations for the bands around the P point are shown in Fig. 2c and its insets. From the dispersion in Fig. 2c, we turn our attention to the WPs between the 39 th and 40 th bands, which have linear dispersions in a large area and are well separated. Second, we have also computed the Chern numbers of the two non-equivalent P points (i.e., P1 and P2). The chiral charges of Weyls at P1 and P2 are computed to be −1 (here, the chiral charge of a Weyl is defined by the Chern number of an enclosed sphere of the lower band). The results of the sphere of the 39 th band around the P point are shown in Fig. 2f. It's consistent with TR symmetry in the system, as mentioned before. Third, as the total chiral charge has to be zero in total, the uncompensated chiral charge of the WPs at two P points suggests that there much be some other nontrivial degenerate point between 39 th and 40 th bands. As a result, a three-fold spin-1 Weyl phonon is found at the H point (highlighted by blue color; only one in the first BZ), formed by the 39 th , 40 th and 41 th bands. The Chern numbers of the spheres of these three bands are computed to be +2, 0, −2, respectively, as shown in Fig. 2c. The results of the sphere of the 39 th band around the H point are shown in Fig. 2g. In other words, the two −1 charged Weyls at the P points are compensated with +2 charged spin-1 Weyl at the H point, shown in Fig. 2e. Then, we emphasize that this system hosts several unconventional properties: (i) The WPs found at the highsymmetry point are of the same chirality at a certain frequency, i.e., the WPs at the P points, which generate the nontrivial Fermi surfaces with nonzero net Chern number. (ii) The nontrivial excitations (quasiparticles) are located at the high-symmetry points, such as the spin-1/2 Weyl at P and the spin-1 Weyl at H, which give rise to the large separation of the compensated topological nodes and large fermi arcs on the terminations. (iii) The spin-1 Weyl phonons are firstly proposed on the boundary of the BZ of a realistic material and is immune to the LOTO modification (see Fig. S1 in Supplementary section A), supporting that it can be observed in experiments. Such spin-1 Weyl phonons at H are also found in K 8 carbon of SG 214 in Supplementary section C.
Exotic surface states Next, we turn to examine the isofrequency surface states To confirm further the topological nature of the two different classes of SEWPs, in Fig. 4a, we plot the isofrequency surface states on the (110) surface (see Fig. 1b), where the high-symmetry points H, P1 and P2 are projected onH,P1 andP2. The evolutions of the surface arcs for three different frequencies are illustrated in Fig. 4b-d, where the double-helicoid surface states and long surface arcs are also clearly visualized. In the range of the three frequencies, a Lifshitz transition occurs. For the frequency of 16.583 THz, the surface arcs are linked to the pair of single and spin-1 WPs, and just terminate on the projection of the P1 point. Moreover, the doublehelicoid surface states in the high-frequency region can also be observed (see Fig. S2 in Supplementary section B).

Discussion
By performing symmetry analyses in 230 SGs, we demonstrate that there are also symmetry-enforced WPs in the spinless systems, such as phononic systems, as the analogs of the TR-enforced Kramers WPs in spinful systems [16]. Such SEWPs can only appears in the nonsymmorphic chiral crystals due to the lack of improper rotation symmetries and the presence of nonsymmorphic symmeties. We summarize the SGs and the corresponding high-symmetry points of SEWPs in Table I. The SEWPs are located on the boundary of the 3D BZ and are protected by nonsymmorphic symmetries, or the combined symmetry of TR and 4 1 . To confirm our findings, we have systematically investigated the phonon spectra of some realistic materials of the SGs in the table by using first-principles calculations. Taking K 2 Sn 2 O 3 of SG 199 as an example, at ∼ 20 THz, the two WPs of the same chirality are found at two P points, compensated with a three-fold spin-1 phonon at the H point. The corresponding surface phonon dispersions display multiple double-helicoid surface states. In addition, the significant separation between the points P and H forms very long and visible phonon surface arcs, which support the stability of the monopole Weyl phonons and the coexisted spin-1 Weyl phonons. Our finds not only present a novel strategy to search for monopole Weyl phonons, but also offer some promising candidates for studying single-Weyl and spin-1 Weyl phonons.

Calculation methods
We carried out the density functional theory (DFT) calculations using the Vienna ab initio Simulation Package (VASP) [37][38][39] with the generalized gradient approximation (GGA) in the form of Pardew-Burke-Ernzerhof (PBE) function for the exchange-correlation potential [40][41][42]. An accurate optimization of structural parameters is employed by minimizing the interionic forces below 10 −6 eV/Å and an energy cut off at 520 eV. The BZ is gridded with 3×3×3 k points. Then the phonon dispersions are gained using the density functional perturbation theory (DFPT), implemented in the Phonopy Package [43]. The force constants are calculated using a 2 × 2 × 2 supercell. To reveal the phonon topological nature, we constructed the phononic Hamiltonian of tight-binding (TB) model and the surface local density of states (LDOS) with the open-source software Wanniertools [44] code and surface Green's functions [45]. Wilson loop method [46,47] is used to find the Chern numbers or topological charge of single Weyl points and spin-1 Weyl points.

A. The phonon dispersion of K2Sn2O3 with LOTO
To illustrate the fact that the three-fold spin-1 Weyl phonon at the high-symmetry point H is immune to the LOTO modification, the phononic dispersions of the material example K 2 Sn 2 O 3 in the presence of the polariton of LOTO splitting are also calculated and shown in Fig. S1. The three bands referring to the 39 th , 40 th and 41 th branches, which form the three-fold spin-1 Weyl phonon with LOTO, are also highlighted in the right panel of Fig. S1. It is clearly seen that the polariton of LOTO splitting does not break the linear crossing bands around the H point, supporting that the three-fold spin-1 Weyl phonon in the present material candidate can be easily observed in experiments. Between the 40 th and 41 th phonon bands of K 2 Sn 2 O 3 as shown in Fig. 2c in the main text, there are only two spin-1 Weyl phonons at the H and Γ points, respectively. The Chiral charge of the Γ point is −2. To show the nontrivial Fermi-arc states associated with them, the phonon surface states on the (110) surface at 18.145 and 18.065 THz are calculated and plotted in Fig. S2b and S2c, respectively. The very long surface arcs and the double helicoid surface states can be clearly seen. In this section, we present some other realistic material candidates in the SGs, which are listed in the Table I in the main text, to illustrate the existence of SEWPs. To the end, we focus our attention on the material examples PNO in SG-24, Ag 3 BiO 3 in SG 80, LiAuO 2 in SG 98, K 2 Pb 2 O 3 in SG 199, SiO 2 in SG 210 and K 6 Carbon in SG 214.
In the following subsections, we first draw their cubic unit cells and the first BZ to show the crystal structures, and then by using symmetry analysis and effective k · p models, we verify the guiding principle of the search of SEWPs as described in the main text. Finally, through calculating their phononic dispersions and DOS, and the Chern numbers of some nontrivial phonon bands to describe the existence of the WPs. The calculation methods are the same as those descried in the main text.

The SEWPs in the material PNO in SG 24
The crystallographic data of PNO are obtained from ref. [1] and the primitive cell is shown in Fig. S3a, where the purple (gray and red) atoms stand for P (N and O) atoms. Each primitive cell contains 6 atoms with two P, two N and two O atoms, and the corresponding BZ is shown in Fig. S3b. PNO belongs to the body-centered cubic structure with the SG I2 1 2 1 2 1 (No. 24).
It is noted that SG 24 hosts the Weyl phonons only at the W point (the high-symmetry points are defined in ref. [5]), and it hosts one irreps of the AG G 7 16 in Table I. This SG has a body-centered cubic Bravais lattice. The operators C 2y and C 2z acting on the primitive lattice vectors (t 1 , t 2 , t 3 ) are described [5] as below: Thus, At the W point ( 1 4 , 1 4 , 1 4 ), the pure translation operator {E| − 1, 0, −1} is expressed as exp[2iπ(−1 + 0 − 1)/4] = −1. Therefore, we get {A, B} = 0, which yields all the phonon bands to be degenerate at the W point.
Next, we analyze a two-band model at the W point in SG 24. We have A 2 = {E|000} = 1, B 2 = {E|000} = 1 with E identity operator and {A, B} = 0. With the matrix representations of A = σ z and B = σ y , the k · p-invariant Hamiltonian is derived as (to the first order), with σ x,y,z Pauli matrices, k x,y,z momentum offset from the P point, and v 1,2,3 real coefficients. Obviously, it's still a Weyl Hamiltonian. The P point of SG 98 and the W point of SG 210 in Table I with anti-unitary commutation relations share the similar results. To confirm the above results from the symmetry analysis, the phononic dispersions and DOS of PNO from the ab initio calculations are illustrated in Fig. S3c. One can see that two-fold Weyl phonons are localized at the highsymmetry point W, one of the corners of the first BZ. Note that the two linear crossing bands possess the Chern number of ±1, indicating its topological nontrivial feature. 3. The SEWPs in the material LiAuO 2 in SG 98 The crystallographic data of LiAuO 2 are obtained from ref. [1] and the primitive cell is shown in Fig. S5a, where the green (yellow and red) atoms stand for Li (Au and O) atoms. LiAuO 2 belongs to the body-centered cubic structure with SG I4 1 22 (No. 98). Each primitive cell contains 8 atoms with 2 Li, 2 Au and 4 O atoms, and the corresponding BZ is shown in Fig. S5b.
SG 98 hosts the Weyl phonons only at the P point (the high-symmetry points are defined in ref. [5]), and it hosts one irreps of the AG G 7 16 in Table I. This SG has a body-centered cubic Bravais lattice. The operators C 2y and C 2z acting on the primitive lattice vectors (t 1 , t 2 , t 3 ) are presented [5] as below: Thus,  4. The SEWPs in the material K 2 Pb 2 O 3 in SG 199 The crystallographic data of K 2 Pb 2 O 3 are obtained from ref. [1] and the primitive cell is shown in Fig. S6a, where the purple (black and red) atoms stand for K (Pb and O) atoms. K 2 Pb 2 O 3 belongs to the body-centered cubic structure with SG I2 1 3 (No. 199), which is in good agreement with previous experimental result [3]. Each primitive cell contains 6 atoms with 4 K, 4 Pb and 6 O atoms, and the corresponding BZ is shown in Fig. S6b.
It is noted that K 2 Pb 2 O 3 belongs to the same SG 199 which has already been studied as an example in the main text, thus the symmetry analysis and the effective k · p model can be referred to the descriptions in the main text. The phonon dispersions and DOS of K 2 Pb 2 O 3 from the ab initio calculations are shown in Fig. S6c. It is clearly seen that two-fold SEWPs and three-fold spin-1 SEWPs are localized at the high-symmetry points P and H, respectively.

The SEWPs in the material SiO 2 in SG 210
The crystallographic data of SiO 2 are obtained from ref. [1] and the primitive cell is shown in Fig. S7a, where the blue (red) atoms stand for Li (O) atoms. SiO 2 belongs to the face-centered cubic structure with SG F4 1 32 (No. 210). Each primitive cell contains 36 atoms including 12 Si, and 24 O atoms, and the corresponding BZ is shown in Fig. S7b.
SG 210 hosts the Weyl phonons only at the W point (the high-symmetry points are defined in ref. [5]) with 2D irrep R10 of AG G 7 16 in Table I. This SG is a face-centered cubic Bravais lattice. The operators C 2x and C 2f acting on the primitive lattice vectors (t 1 , t 2 , t 3 ) are presented [5] as below: Thus, where, C 2d is the Jone's symbol defined in cubic lattice as It's easy to prove that C 2x C 2y =C 2y C 2x =C 2d . At the W point ( 1 2 , − 1 4 , 3 4 ), the pure translation operator {E|1, 0, 0} is expressed as exp[2iπ/2] = −1. Therefore, we get {A, B} = 0, which yields all the phonon bands to be degenerate at the P point. We have checked that there is no symmetry-protected degeneracy on the high-symmetry planes/lines that cross the W point. Especially, we have to exclude those contain inversion symmetry and improper rotation symmetries.
The phonon dispersions and DOS of SiO 2 from the ab initio calculations are shown in Fig. S7c. It is clearly seen that the two-fold SEWPs are located at the high-symmetry point W, which is in agreement with the above results from the symmetry analysis. It is noted that the phononic dispersion of SiO 2 in SG 210 shows much imaginary frequencies, because it isn't a very stable crystal. 6. The SEWPs in the material K 6 Carbon in SG 214 The primitive cell of K 6 Carbon in SG 214 is described in Fig. S8a, where the brown atoms stand for C atoms. K 6 Carbon belongs to the body-centered cubic structure with SG I4 1 32 (No. 214), which is in good agreement with previous calculation result [4]. Each primitive cell contains 6 C atoms, and the corresponding BZ is shown in Fig. S8b.
SG 214 same as SG 199 hosts the Weyl phonons only at the P point (the high-symmetry points are defined in ref. [5]), even though it hosts three different irreps of AG G 3 48 in Table I. Since the exactly same position and symmetry of the P point in SG 214 and SG 199, we can easily get the same anti-commutation relation {A, B} = 0 by same process where A:{C 2x | 1 2 1 2 0} and B:{C 2z | 3 2 1 1 2 }, which yields all the phonon bands to be degenerate at the P point. We have checked that there is no symmetry-protected degeneracy on the high-symmetry planes/lines that cross the P point. Especially, we have to exclude those contain inversion symmetry and improper rotation symmetries.
The phonon dispersions and DOS of K 6 Carbon from the ab initio calculations are shown in Fig. S8c. It is clearly seen that two-fold SEWPs and three-fold spin-1 SEWPs are localized at the high-symmetry points P and H, respectively.