Abstract
In spinful electronic systems, timereversal symmetry makes that all Kramers pairs at the timereversalinvariant momenta are Weyl points (WPs) in chiral crystals. Here, we find that such symmetryenforced WPs can also emerge in bosonic systems (e.g. phonons and photons) due to nonsymmorphic symmetries. We demonstrate that for some nonsymmorphic chiral space groups, several highsymmetry kpoints can host only WPs in the phononic systems, dubbed symmetryenforced Weyl phonons (SEWPs). The SEWPs, enumerated in Table 1, are pinned at the boundary of the threedimensional (3D) Brillouin zone (BZ) and protected by nonsymmorphic crystal symmetries. By performing firstprinciples calculations and symmetry analysis, we propose that as an example of SEWPs, the twofold degeneracies at P are monopole WPs in K_{2}Sn_{2}O_{3} with space group 199. The two WPs of the same chirality at two nonequivalent P points are related by timereversal symmetry. In particular, at ~17.5 THz, a spin1 Weyl phonon is also found at H, since two Weyl phonons at P carrying a nonzero net Chern number cannot exist alone in the 3D BZ. The significant separation between P and H points makes the surface arcs long and clearly visible. Our findings not only present an effective way to search for WPs in bosonic systems, but also offer some promising candidates for studying monopole Weyl and spin1 Weyl phonons in realistic materials.
Introduction
Topological phonons^{1,2,3,4,5,6,7,8}, referring to the quantized excited vibrational states of interacting atoms, have most recently attracted attention in condensed matter physics because of their unique physical nature^{8,9,10,11,12,13}. In similarity to various quasiparticles in electronic systems, topological phonons, such as (spin1/2 or monopole) Weyl, Dirac, spin1 Weyl, and charge2 Dirac phonons, have been predicted/observed in 3D momentum space of solid crystals^{14,15,16,17,18,19,20,21,22}, strengthening largely our understanding of elementary particles in the universe. For instance, Zhang et al.^{22} predicted that both spin1 Weyl phonons and charge2 Dirac phonons exist in the CoSi system. The coexistence of the above two different classes of topological phononic quasiparticles exhibits exotic topologically nontrivial features, such as noncontractible surface arcs and doublehelicoid surface states^{23}. 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 surfaces in materials. The phononic systems with these particular properties provide a good platform for studying topological bosonic states in experiments.
According to NielsenNinomiya nogo theorem^{24}, WPs always come in pairs with opposite chirality, acting as sources/sinks of Berry curvature in the 3D BZ^{25}. The WPs are formed by two bands with linear dispersions in 3D momentum space, which have Chern numbers (C) of ±1 and robust against small perturbations. They are usually not easy to be predicted due to the lack of symmetry protections, and the predictions of WPs usually require comprehensive numerical calculations in the 3D BZ^{26}. However, in the spinful electronic systems, Kramers pairs at the timereversalinvariant momenta are enforced to be WPs by timereversal (TR) symmetry in chiral crystals (Fig. 1a), where there are no improper rotation symmetries, such as inversion or mirror^{15,16}. In this work, by checking the symmetries of 230 space groups (SGs), we have uncovered that symmetryenforced WPs can also emerge in bosonic systems, such as phonons (mainly discussed in the work), photons, and so on, due to the presence of nonsymmorphic symmetries (Fig. 1b, c). These WPs are all pinned at the highsymmetry kpoints on the boundary of the 3D BZ. Unlike the TRenforced WPs in spinful electronic systems^{15}, where the WPs are usually buried in bulk states due to the weak strength of spin−orbit coupling, the nonsymmorphiccrystalsymmetryenforced Weyl phonons can be well exposed. Consequently, the associated surface arcs are long and robust, which can be easily probed in future experiments. By performing symmetry analysis in 230 SGs in the presence of TR symmetry, we have demonstrated that for some chiral SGs, several highsymmetry kpoints can host only WPs in the phononic systems, dubbed symmetryenforced Weyl phonons (SEWPs). We enumerate all the SEWPs at the highsymmetry kpoints of the SGs in Table 1. In this table, all the phonon bands are doubly degenerate at those highsymmetry kpoints and each twofold degeneracy represents a WP. Thus, one can easily predict WPs in such systems as long as the materials are of the SGs in Table 1. The results significantly lower the difficulty to predict the WPs in bosonic systems.
By employing firstprinciples calculations, we predicted that as an example of the SEWPs, twofold degeneracies at the P point 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 timereversal symmetry. Therefore, the WPs at two P points host the same chirality. Second, at ~17.5 THz, a spin1 Weyl phonon is also found at H, since two Weyl phonons at P carrying a nonzero net Chern number cannot exist alone in the 3D BZ. Third, the spin1 Weyl phonon here has been found to locate on the boundary of the BZ, which is robust against the LOTO (longitudinal and transverse optical phonon splitting) modification in the phonon spectrum. Lastly, the symmetryrelated WPs host the same chiral charge, giving rise to nontrivial isofrequency surfaces of phonons, associated with nonzero Chern numbers. In addition, the long surface arcs and the doublehelicoid states are presented as well. More examples of SEWPs can be found in the Supplementary Information. These new findings not only provide an effective way to search for monopole WPs in bosonic systems, but also predict some promising candidates for studying topological quasiparticles in experiments.
Results
Searching for SEWPs by symmetry analysis
The guiding principle of our search is to find twodimensional irreducible representations (irreps) of the (little) group of lattice symmetries at highsymmetry kpoints in the 3D BZ for each of the 230 SGs in the presence of TR symmetry; the dimension of the irreps corresponds to the number of bands that meet at the highsymmetry kpoints. Then, one has to exclude the SGs that contain improper rotational symmetries or twofold screw symmetries (\({\bar{C}}_{2}\)) with \({[{\bar{C}}_{2}{\mathcal{T}}]}^{2}=1\), to make sure that there is no double degeneracy along any highsymmetry line/plane crossing these kpoints. Since we are interested in bosonic systems, we consider only the singlevalued representations; TR symmetry is an antiunitary that squares to 1 (i.e., \({[{\mathcal{T}}]}^{2}=1\)). We find that the twofold WPs pinned at highsymmetry kpoints in bosonic systems can have Chern numbers of ±1 [i.e. (monopole) WPs], ±2 [i.e., double WPs] and ±4 [i.e., quadruple WPs], respectively. The (monopole) WPs are formed by two bands with linear dispersions^{27} and can be stabilized at general points in 3D momentum space, while two bands of the double and quadruple WPs have nonlinear dispersions^{27,28,29,30,31,32,33}, which are usually pinned at highsymmetry kpoints by symmetries. A complete list of the highsymmetry kpoints, where these three kinds of twofold WPs can be pinned and protected in bosonic systems, is presented in Table 1 in Section A of the Supplementary Information. For most of them, besides the twodimensional irreps, onedimensional and/or threedimensional irreps are also allowed. However, at several specific kpoints, only the twodimensional irrep of WPs is allowed in the phononic systems, dubbed SEWPs, which are mainly focused on in this work.
The results of SEWPs are summarized in Table 1. These SEWPs are monopole WPs and are located on the boundary of the 3D BZ. All of them are chiral SGs with nonsymmorphic symmetries, and all representations are projective; these are in fact necessary ingredients for the (spin1/2) Weyl excitations in the phononic systems. We find that most of twofold degeneracies are protected by the anticommutation relation of two unitary operators (i.e., {A, B} = 0) in Table 1, except for SG 80. At the P point of SG 80, an antiunitary symmetry of its little group is the combined operator of time reversal (\({\mathcal{T}}\)) and fourfold screw symmetry (\({{\bf{4}}}_{1}\equiv \{{C}_{4z} \frac{3}{4}\frac{1}{4}\frac{1}{2}\}\)) (see the definition in ref. ^{34}. One can check that \({({\mathcal{T}}\cdot {{\bf{4}}}_{1})}^{4}=\{E 110\}={e}^{2i\pi (\frac{1}{4}+\frac{1}{4})}=1\), which enforces a Kramerslike degeneracy as discussed in ref. ^{35}.
Then, we take SG 199 as an example to illustrate the anticommutation relation in the main text (see more derivations for all other SGs in Table 1 in Section D of the Supplementary Materials). SG 199 hosts only Weyl phonons at the P point (the highsymmetry points are defined in ref. ^{34}), even though it hosts three different irreps of the AG \({G}_{48}^{3}\) in Table 1. This SG has a bodycentered cubic Bravais lattice. The operators C_{2x} and C_{2z} acting on the primitive lattice vectors (t_{1}, t_{2}, t_{3}) are presented^{34} as follows:
Thus,
At the P point (\(\frac{1}{4},\frac{1}{4},\frac{1}{4}\)), the pure translation operator {E∣3, 2, 1} is expressed as e^{2iπ(3+2+1)/4} = −1. Therefore, we get {A, B} = 0, which yields all the phonon bands to be at least twofold degenerate at the P point. Note that no higher ndimensional irreps (n > 2) are found at P. In addition, we have checked that there is no symmetryprotected degeneracy on the highsymmetry planes/lines crossing the P point.
Effective k ⋅ p models
Let us consider a twoband model at the P point in SG 199 first. We have A^{2} = {E∣000} = 1, B^{2} = {E∣220} = 1 with E the identity operator and {A, B} = 0. With the matrix representations of A = σ_{x} and B = σ_{z}, 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 P point, and v_{1,2,3} real coefficients. Obviously, it is a Weyl Hamiltonian. Other SGs in Table 1 with antiunitary commutation relations share the similar results. We notice that the P point is a TR noninvariant point at a corner of the BZ (i.e., P ≠ −P). Those systems host another Weyl phonon of the same chirality at −P due to TR symmetry, indicating that there must be some other nontrivial excitation(s) in the systems, since the two Weyl phonons carrying a nonzero net Chern number cannot exist alone in the 3D BZ. Here, we choose K_{2}Sn_{2}O_{3} in SG 199 as an example for illustration, which exhibits a spin1 Weyl phonon (C = 2) at H and two Weyl phonons (C = −1) at P between the 39th and 40th bands.
Then we consider the P point in SG 80, where we cannot find two symmetry operators with the anticommutation relation. We consider two symmetry operators at P point: \(D=\{{C}_{2z} 100\}={{\bf{4}}}_{1}^{2}\) and \({\mathcal{T}}\cdot {{\bf{4}}}_{1}\), where 4_{1} is a nonsymmorphic fourfold rotational symmetry, followed by a fractional lattice translation [T(\(\overrightarrow{c}\)/4), where \(\overrightarrow{c}\) is a lattice constant in the z direction]. It is worth noting that 4_{1} is not a symmetry operator that keeps P invariant (see more details in the Supplementary Information). We can express the two operators as D = −iσ_{z} and \({\mathcal{T}}\cdot {{\bf{4}}}_{1}=\frac{1}{\sqrt{2}}({\sigma }_{x}+{\sigma }_{y}){\mathcal{K}}\), which meet the conditions: \({({\mathcal{T}}\cdot {{\bf{4}}}_{1})}^{2}=D\) and \({({\mathcal{T}}\cdot {{\bf{4}}}_{1})}^{4}=1\). Thus, the k ⋅ p invariant Hamiltonian is derived as (to the first order of k),
with \(\theta =\frac{{v}_{2}}{{v}_{1}}\), \({k}_{  }=\frac{{k}_{x}\,+\,\theta {k}_{y}}{\sqrt{1\,+\,{\theta }^{2}}}\) and \({k}_{\perp }=\frac{\theta {k}_{x}\,\,{k}_{y}}{\sqrt{1\,+\,{\theta }^{2}}}\). It is also a Weyl Hamiltonian with isotropy in the k_{x}−k_{y} plane.
SEWPs in realistic materials
Based on our symmetry analysis, the phonon dispersions of any material in the SGs in Table 1 have to contain WPs at those highsymmetry points. To confirm the theoretical results, we have systematically performed the ab initio phonon calculations on some materials for each SG in Table 1. 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 the discussions for other SGs in the Supplementary Information. The crystallographic data of K_{2}Sn_{2}O_{3} are adopted from ref. ^{36}, 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 bodycentered cubic structure with SG I2_{1}3. Each primitive cell contains 14 atoms with four K, and four Sn and six O atoms. The bulk BZ, (001) surface BZ and (110) surface BZs are shown in Fig. 2b.
The calculated phonon dispersions of K_{2}Sn_{2}O_{3} are shown in Fig. 2c. It is clearly seen that there are some band crossings (degeneracies) at highsymmetry kpoints, especially for the optical dispersions. First, we do find that all the phonon bands are doubly degenerate at P point, resembling monopole WPs. The corresponding results of the Chern number calculations for the bands around P point are shown in Fig. 2c and its insets. From the dispersions in Fig. 2c, we turn our attention to the WPs formed by the 39th and 40th bands, which have linear dispersions in a wide frequency range. Second, we have also computed the Chern numbers of the two nonequivalent P points (i.e., P1 and P2). The chiral charges of WPs at P1 and P2 are computed to be −1. Here, the chiral charge of a WP is defined by the Chern number of the lower band, which is computed by the Wilson loop technique on a sphere enclosing the WP^{37,38}. The results of the sphere for the 39th band around the P point are shown in Fig. 2g. It is consistent with TR symmetry in the system, as mentioned before. By fitting the two phonon bands in the vicinity of P point, the v_{1}, v_{2} and v_{3} coefficients in Eq. (3) are given as 2.19, 2.19 and −2.19 THz ⋅ Å. Third, since two Weyl phonons carrying a nonzero net Chern number cannot exist alone in the 3D BZ, a threefold spin1 Weyl phonon is found at H point (highlighted by blue color), formed by the 39th, 40th, and 41st bands. The Chern numbers of these three bands are computed to be +2, 0, −2, respectively, as shown in Fig. 2c. The results of the sphere for the 39th band around the H point are shown in Fig. 2f. As illustrated in Fig. 2h, the spin1 WP with C = +2 at H point acts as the “source” point, whereas two monopole WPs with C = −1 at P points can be viewed as the “sink” points of the Berry curvature.
Consequently, this system hosts several unconventional properties: (i) The symmetryrelated WPs are of the same chirality, i.e., the monopole WPs at the P points, which can generate the nontrivial isofrequency surfaces of photon with nonzero Chern numbers. (ii) The nontrivial excitations (quasiparticles) are pinned at the highsymmetry kpoints, such as the monopole Weyl phonons at P and the threefold spin1 Weyl phonon at H, which give rise to large surface arcs due to the large separation of the sources and sinks of the Berry curvature. (iii) The spin1 Weyl phonon is proposed on the boundary of the BZ of a realistic material and is robust against the LOTO modification (see Supplementary Fig. S1 in the Supplementary Information), supporting that it could be observed in future experiments. Such a spin1 Weyl phonon at H is also found in K_{8} carbon of SG 214 in Section D of the Supplementary Information.
Exotic surface states
Then, we turn to examine the isofrequency surface contours and the surface arcs to explore the exotic physical behaviors of the SEWPs. As the two P points are projected to the BZ corner (\(\widetilde{{\rm{M}}}\)) and the H point to the BZ center (\(\widetilde{\Gamma }\)) in Fig. 3a, we plot the (001)surface phonon dispersions along the green line of Fig. 3a (i.e., \(\widetilde{{\rm{M}}}\)\(\widetilde{\Gamma }\)\(\widetilde{{\rm{M}}}\)) in Fig. 3b. At two chosen frequencies of 16.58 (i.e., the frequency of the WPs at P) and 16.73 THz, the arclike surface states are illustrated in Fig. 3a, c, respectively. The surface arcs connect the two monopole WPs at \(\widetilde{{\rm{M}}}\) and one spin1 Weyl phonon at \(\widetilde{\Gamma }\). As expected, the isofrequency surface contours display doublehelicoid states^{39}, which are spiral clockwise with increasing frequency. The surface phonon dispersions along the squareshaped path around \(\widetilde{{\rm{M}}}\) show doublehelicoid surface states (see Fig. 3d), which verifies further the topological nontrivial feature of the two monopole WPs at P points.
Next, we plot the (110)surface phonon dispersions in Fig. 4a and the isofrequency surface contours in Fig. 4b−d. On the (110)surface BZ (Fig. 4b), the H, P1 and P2 points are projected to \(\overline{{\rm{H}}}\), \(\overline{{\rm{P}}}1\) and \(\overline{{\rm{P}}}2\), respectively. The evolution of the surface arcs is illustrated by isofrequency surface contours for three frequencies, in which the surfaces arcs are clearly visualized. In this frequency range, a Lifshitz transition occurs clearly in the arclike surface states. Moreover, the surfaces arcs at higher frequencies are presented in Supplementary Fig. S2 of the Supplementary Information.
Discussion
By performing symmetry analysis in 230 SGs in the presence of TR symmetry, we demonstrate that there are also symmetryenforced WPs in the bosonic systems, e.g. phonons. We have given a complete list of highsymmetry kpoints, where twofold Weyl nodes (e.g. C = ±1, ±2, and ±4) are protected. This list can guide future experiments in the study of various WPs. Among them, several kpoints can support only twofold Weyl phonons, dubbed SEWPs. Such SEWPs have Chern numbers of ±1 and appear in the nonsymmorphic chiral crystals due to the lack of improper rotation symmetries and the presence of nonsymmorphic symmetries. We summarize the highsymmetry kpoints of SEWPs in Table 1. The SEWPs are pinned at the boundary of the 3D BZ 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 Table 1 by using firstprinciples calculations. Taking K_{2}Sn_{2}O_{3} of SG 199 as an example, two Weyl phonons at P and a spin1 Weyl phonon at H appear together in between the 39th and 40th bands. The corresponding surface phonon dispersions display multiple doublehelicoid surface states. In addition, the significant separation between the points P and H forms very long and visible surface arcs. Our findings not only present an effective way to search for WPs in bosonic systems, but also offer some promising candidates for studying monopole Weyl and spin1 Weyl phonons.
Methods
We carried out the density functional theory calculations using the Vienna ab initio Simulation Package^{40,41,42} with the generalized gradient approximation in the form of Pardew−Burke−Ernzerhof function for the exchangecorrelation potential^{43,44,45}. An accurate optimization of structural parameters is employed by minimizing the interionic forces below 10^{−6} eV/Å and an energy cutoff at 520 eV. The BZ is gridded with 3 × 3 × 3 kpoints. Then the phonon dispersions are gained using the density functional perturbation theory, implemented in the Phonopy Package^{46}. The force constants are calculated using a 2 × 2 × 2 supercell. To reveal the phonon topological nature, we constructed the phononic Hamiltonian of the tightbinding (TB) model and obtained the surface local density of states with the opensource software Wanniertools^{47} code and surface Green’s functions^{48}. The irreps of the phonon bands can be computed by the program—ir2tb—on the phononic Hamiltonian of the TB model^{49}. Wilson loop method^{37,38} is used to find the Chern numbers or topological charge of monopole WPs and spin1 WPs.
Data availability
Data are available from the authors upon reasonable request.
Code availability
The related codes are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work is supported by the National Science Foundation of China (Grants No. 11774104, 11504117, and 11274128) and the Strategic Priority Research Program of Chinese Academy of Sciences (XDB33000000).
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Z.W. and H.H.F. conceived and supervised the project. Q.B.L. and H.H.F. did the phonon calculations. Y.Q. and Z.W. did the symmetry analysis. All authors contributed to analyzing the results and writing the manuscript.
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Liu, Q., Qian, Y., Fu, H. et al. Symmetryenforced Weyl phonons. npj Comput Mater 6, 95 (2020). https://doi.org/10.1038/s41524020003588
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