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REVIEW 3 major objections 5 minor 77 references

Catalog of phonon emergent particles

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Pith's one-line read This paper establishes a complete mapping from Wyckoff positions to phonon emergent particles at high-symmetry points in all 230 space groups, and uses it to identify 20,516,167 such particles in 111,872 materials.

desk verdict A parameter-free symmetry criterion for phonon EMPs across 111,872 materials, with a real but contained verification gap around the external EMP taxonomy. read the letter →

arxiv 2411.15840 v1 pith:FCTHIHH4 submitted 2024-11-24 cond-mat.mtrl-sci

Dongze Fan , Hoi Chun Po , Xiangang Wan , Feng Tang This is my paper · ORCID
classification cond-mat.mtrl-sci PACS 63.20.-e
keywords phononemergentparticlesWyckoffpositionsspacegroupshigh-symmetrypointstopologicalphononsangularmomentumirreduciblerepresentationshigh-throughputcatalog
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that the phonon emergent particles at a crystal's high-symmetry points—band crossings whose low-energy excitations behave like Weyl, Dirac, triple, sextuple, and related particles—are fixed by symmetry alone. The authors build a complete map from the occupied Wyckoff positions to the little-group irreducible representations of the phonon modes at each high-symmetry point in all 230 space groups. From that map they prove that, with one exception, every emergent particle type that is symmetry-allowed at a high-symmetry point is guaranteed to appear in any structure of that space group. They also trace which atomic sites carry phonon angular momentum, and apply the machinery to produce a catalog of 20,516,167 emergent particles in 111,872 materials, with over 90% of those materials hosting at least one.

What carries the argument

The central object is the Wyckoff-position occupancy count $n^{w}_{k,j}$: when atoms occupy Wyckoff position $w$, this number gives the occurrences of little-group (co-)irrep $j$ at high-symmetry point $k$ in the phonon spectrum. Because phonon vibrations transform as three vector degrees of freedom, only vector representations enter, turning the problem into a finite enumeration over all 230 space groups. This count supplies both the complete mapping from Wyckoff positions to emergent particles and the atomic contributions to phonon angular momentum. Two enforcement rules are derived from it: in the first case every Wyckoff position in the space group contributes the relevant irrep, so the emergent particle is unavoidable; in the second case the Wyckoff positions that do not contribute would force a different, higher-symmetry space group, so the emergent particle still must appear.

What would settle it

Take any compound whose space group and Wyckoff occupations match an entry in Extended Data Table 1 or 2, compute its phonon band structure at the listed high-symmetry point with converged force constants and a dense q-mesh, and check for the predicted irrep content and degeneracy; a single absence of a Wyckoff-enforced emergent particle would falsify the catalog, as would a structure that stays in the same space group while avoiding the predicted crossing.

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Extended reading notes

Core claim

The central claim is that phonon wavefunctions at high-symmetry points are symmetry-determined far more completely than previously exploited: the number of times each little-group irrep appears is fixed by the occupied Wyckoff positions, and because phonon displacements behave like three p-like vector orbitals, the full mapping from Wyckoff positions to irrep counts can be enumerated for every one of the 230 space groups. Combining this with the known correspondence between irreps and the 19 emergent particle types, the authors obtain a complete Wyckoff-position-to-emergent-particle mapping. Exhaustively checking all Wyckoff positions, they find that apart from P-WNLs (nodal points sitting on Weyl nodal lines), all emergent particles at high-symmetry points are enforced to definitely appear; P-WNLs can fail to appear only in space groups 216, 221, 225, 227, and 229 for particular Wyckoff occupations. The same symmetry-determined wavefunctions also show that phonon modes with nonzero angular momentum at high-symmetry points exist only in 20 noncentrosymmetric space groups, and identify which Wyckoff positions contribute to that angular momentum.

Load-bearing premise

The catalog is only as reliable as the external classification of the 19 emergent particle types and their mapping to little-group irreps, and, for the database scan, the assigned space groups and standardized atomic positions; an error in either propagates into every enforcement rule and material count.

Editorial extensions

If this is right

  • Materials screening can be done on crystal structure alone: knowing only the space group and Wyckoff occupations, one can list every high-symmetry-point emergent particle without performing a phonon calculation.
  • Because over 90% of the 111,872 catalogued materials host at least one emergent particle, topological phonon candidates are abundant rather than exceptional; the 2,309 nearly ideal candidates with well-separated, all-positive-frequency bands are the most accessible subset.
  • For any emergent particle, the contributing atoms are identified by Wyckoff position, so site substitution and isotope engineering can be aimed at the carriers of the crossing or of the phonon angular momentum.
  • The 20 noncentrosymmetric space groups with nonzero high-symmetry-point phonon angular momentum define the structural universe for chiral-phonon physics at high-symmetry points.
  • Only five space groups can avoid P-WNLs at high-symmetry points, and only for specific Wyckoff occupations; any material in those space groups must be checked before assuming a Weyl-nodal-line point exists.
  • The symmetry-determined wavefunctions supply the irrep, frequency, degeneracy, and topological character of every high-symmetry-point level for the 10,034 fully computed materials, giving a checkable reference against first-principles phonon data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Wyckoff-position-to-irrep bookkeeping should transfer to any finite-dimensional excitation space, such as photon modes or electron bands built from fixed orbitals; the paper notes this transferability, and deriving the explicit enforcement rules for electrons would be a direct next step.
  • The near-universal enforcement of high-symmetry-point emergent particles suggests that many reported parameter-dependent or seemingly accidental phonon crossings are actually unavoidable once the atomic sites are fixed, so recalculating with stricter parameters should confirm them rather than remove them.
  • The 2,309 nearly ideal candidate materials provide a natural experimental test: measured phonon spectra at the predicted high-symmetry points should show the isolated degeneracies, and any discrepancy would point to force-constant errors rather than symmetry errors.
  • Because the angular-momentum-contributing Wyckoff positions are explicitly listed, one could design alloys or layered exfoliation precursors that place heavy or magnetically active atoms on those sites to amplify chiral-phonon effects.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper constructs a group-theoretic mapping between Wyckoff positions (WYPOs) in the 230 space groups and the irreducible representations (irreps) of phonon modes at high-symmetry points (HSPs). Starting from the observation that phonon displacement modes transform as polar vectors, the authors compute the number of times each little-group irrep occurs at every HSP when a given WYPO is occupied. From this mapping they derive rules for when emergent particles (EMPs) are enforced to appear, with the central claim that, apart from P-WNL points, every EMP type allowed at HSPs is guaranteed to appear once the space group and WYPO occupancies are fixed. They apply the mapping to 10,034 PhononDB and 101,838 ICSD materials to report 20,516,167 phonon EMPs, and they use the same symmetry data to identify WYPO contributions to phonon angular momentum in 20 noncentrosymmetric space groups. Four illustrative materials are analyzed with first-principles phonon spectra, and the irrep multiplicities from symmetry analysis are stated to be exactly consistent with first-principles results for the PhononDB set.

Significance. If the central mapping is correct, the paper provides a parameter-free, structure-only route to predicting the occurrence and atomic origin of phonon emergent particles at high-symmetry points, going beyond the irrep-multiplicity statistics used in earlier topological-phonon catalogs. The enumeration over all WYPOs in all 230 space groups, together with the exact consistency on 10,034 first-principles calculations, would make this a valuable reference resource for topological phonons and for phonon angular momentum. The work also carries a genuinely useful catalog: searchable tables of materials, EMP types, frequencies, and WYPO-resolved contributions, including a set of candidate materials for experiments. The main strength is that the core mapping is a derivation with no fitted parameters and is checked by an independent first-principles calculation.

major comments (3)
  1. [Section III and Extended Data Table 1] The load-bearing exhaustiveness claim, stated in Section III as 'other than P-WNLs, all EMPs at HSPs are enforced to definitely appear through exhaustively examining all WYPOs in the 230 SGs', is inherited from the external classification of Ref. [60]: the list of 19 EMP types and the correspondence between little-group irreps and EMP types are taken as input (Extended Data Table 1 and Section II). The validation on PhononDB materials checks only the number of occurrences of each irrep, not whether the EMP-type labeling assigned by the irrep-to-EMP correspondence is correct. A missing EMP type or a mis-assigned irrep-to-EMP entry would invalidate the 'all EMPs' and 'definitely appear' statements for the affected space groups and k-points. Please provide an independent check of the taxonomy, for example by constructing explicit k·p models for a representative subset of SG/HSP entries, or by cross-validating the irrep-to-EMP assignment against another classification, and state this external input as an explicit assumption in the main text.
  2. [Section III and Extended Data Table 2] The two WYPO-enforcement rules and the exceptional P-WNL list are stated as results of an 'exhaustively examining all WYPOs in the 230 SGs', but no algorithm, proof, or computer code is supplied to support the completeness of that examination. The Supplemental Materials provide the resulting tables, which are necessary but not sufficient for a reader to verify the enumeration or to extend it. In particular, rule (2) relies on the statement that when atoms occupy only the non-contributing WYPOs, the resulting structure belongs to a different space group; this is a nontrivial group-theoretic and crystallographic assertion. Please provide the enumeration script or an explicit mathematical argument for how all WYPO combinations and resulting space groups were tested, so that the 230-SG completeness claim is reproducible.
  3. [Section V and Extended Data Eq. 1] The phonon angular momentum results in Section V and Table I require a stated basis convention for degenerate modes. The formula for phonon AM in Extended Data Eq. 1 is evaluated on individual phonon eigenvectors, but for a degenerate irrep the eigenvectors are defined only up to a unitary rotation within the degenerate subspace, and the per-branch AM is not invariant under such rotations. The paper should specify the convention used (for example, diagonalization of a symmetry operator commuting with the dynamical matrix, or a trace over the degenerate subspace) and justify that the listed WYPO contributions to nonzero phonon AM are basis-independent. As written, the claim that a WYPO 'contributes' to the AM of a particular degenerate branch is not uniquely defined.
minor comments (5)
  1. [Section III and Extended Data Table 2] The notation 'P-WNLs' is used in two senses: as the plural of P-WNL and as the distinct EMP type P-WNLs. The sentence 'only for SGs 216, 221, 225, 227 and 229, P-WNLs have a chance to be not present' is confusing because Extended Data Table 2 contains separate rows for P-WNL and P-WNLs, and the listed exceptional SGs do not match the SG sets in those rows. Please disambiguate the notation, for example by writing 'P-WNL(s)' for plural occurrences and 'P-WNLs' only for the distinct type, and reconcile the exceptional-SG statement with the table entries.
  2. [Extended Data Table 3] The header 'The list of MIPD for layered materials' contains a typo: it should be 'MPID' (Materials Project ID). The same abbreviation is introduced in Section II as 'MPID'.
  3. [Extended Data section 1] The use of plural forms such as 'P-NS[s]' and 'P-WNLss' is unconventional and makes the text harder to read. A single sentence defining the plural conventions at first use would help, and the paper should consistently use those conventions throughout.
  4. [Section II and Supplemental Material] The Supplemental Material is hosted at an institutional box URL (https://box.nju.edu.cn/d/132f8ac8aac545f2862e/), which is not a permanent archival link. For a catalog paper, a stable repository with a DOI would be preferable to ensure long-term accessibility of the large data tables.
  5. [Abstract] The phrase 'unambiguously identify 20,516,167 phonon EMPs' overstates the certainty for the ICSD subset, where EMPs are assigned from space-group and WYPO information alone without phonon frequency calculations. It would be more precise to say 'predicted' for structure-only assignments and to state explicitly that the first-principles frequency-level identification is performed only for the 10,034 PhononDB materials.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the WYPO-to-irrep mapping is a symmetry-theoretic derivation independently checked against first-principles phonon calculations; the EMP taxonomy is an external input, not an output of the fit.

full rationale

The paper's central derivation is a group-theoretic decomposition, not a fit. It obtains n^w_{k,j} by decomposing the phonon representation induced from each Wyckoff position into little-group irreps at each high-symmetry point across the 230 space groups; this is a self-contained symmetry calculation whose only inputs are the space-group/Wyckoff data and the standard phonon vector representation. The EMP taxonomy and irrep-to-EMP assignments are taken from Ref. [60], an external classification by a different set of authors; using it as an input is not circular, because the paper's own contribution—the WYPO-to-irrep multiplicity map and the enforcement rules—does not presuppose the EMP labels it is used to derive. The validation against 10,034 PhononDB materials compares the symmetry-derived irrep multiplicities with those obtained from first-principles force constants and finds complete consistency, which is an independent benchmark rather than a fitted parameter. The statement that 'other than P-WNLs, all EMPs at HSPs are enforced to definitely appear' inherits the completeness of Ref. [60], but this is a correctness/verification dependence on an external, independently published classification, not a circular reduction: at no point does the paper define an EMP in terms of the WYPO counts, fit a parameter to counts and then 'predict' counts, or rely on a load-bearing self-citation. No circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central result is a symmetry decomposition with no fitted parameters and no newly postulated physical entities. The emergent particle names (Weyl, Dirac, etc.) are taken from prior literature.

assumptions (4)
  • domain assumption Phonon modes transform under the vector representation of the site-symmetry group for each atom.
    The paper explicitly states that the three Cartesian displacement directions per atom are treated like three p orbitals, so only the vector representation enters (Fig. 1 caption and Section I). This is standard for phonons at high-symmetry points in the harmonic approximation.
  • domain assumption The EMP classification and the irrep-to-EMP correspondence of Ref. [60] are complete and correct.
    Used as the external input for mapping irreps to the 19 EMP types (Section III and Extended Data Table 1). The present paper does not re-derive this classification.
  • domain assumption The space-group and Wyckoff-position labels in PhononDB@kyoto-u and ICSD are correct for the collected structures.
    The catalog depends on these assignments; 44 non-conformant PhononDB structures were re-standardized with phonopy and recalculated (Extended Data section 1).
  • standard math Induced representations from site-symmetry groups give the full phonon representation at a given k-point.
    The WYPO-to-irrep counting rests on standard group-theoretic induction, implicitly invoked throughout Section III and the SM tables.

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Cite this review

Pith. "Pith review of Catalog of phonon emergent particles." pith.science (2026). https://pith.science/paper/FCTHIHH4

@misc{pith2026241115840,
  author       = {Pith},
  title        = {Pith review of: Catalog of phonon emergent particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FCTHIHH4}},
  note         = {Machine review of arXiv:2411.15840}
}
read the original abstract

The outcome of conventional topological materials prediction scheme could sensitively depend on first-principles calculations parameters. Symmetry, as a powerful tool, has been exploited to enhance the reliability of predictions. Here, we establish the relationship between the Wyckoff positions (WYPOs) and the phonon wavefunctions at each high-symmetry point (HSP) in all 230 space groups (SGs). Based on this, on one hand, we obtain a complete mapping from WYPO to the occurrence of emergent particles (EMPs) at each HSP in 230 SGs, and establish several rules of enforcing EMPs for phonons; on the other hand, we determine the contribution of the WYPO to the phonon angular momentum. Then we unambiguously identify 20,516,167 phonon EMPs in 111,872 materials in two databases. The purely symmetry-determined wavefunctions generalize the conventional Bloch theorem, could find a wide scope of application to physical properties related with basis functions of irreducible representations.

Figures

Figures reproduced from arXiv: 2411.15840 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram summarizing the results obtained in this work. The locations of these results are indicated in the orange boxes. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The number of different types of EMPs, which we identified in the phonon spectra of materials from PhononDB@kyoto-u and ICSD, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The phonon spectra of four ideal topological phonon materials. (a) The phonon dispersion of Na [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Works this paper leans on

77 extracted references · 56 canonical work pages

  1. [60]

    Z.-M. Yu, Z. Zhang, G.-B. Liu, W. Wu, X.-P. Li, R.-W. Zhang, S. A. Yang, and Y . Yao, Encyclopedia of emergent particles in three-dimensional crystals, Sci. Bull. 67, 375 (2022)

    2022

  2. [1]

    1) where ξx i,kν is the x-component of the phonon eigenvector with wave vector k and branch ν for the atom i, and the sum- mation is performed over all atoms within the unit cell

    Details of calculation process For the calculation of phonon AM, the formula along the z direction is as follows [47]: lz kν = 2ℏ X i Im(ξx∗ i,kνξy i,kν), (Extended Data Eq. 1) where ξx i,kν is the x-component of the phonon eigenvector with wave vector k and branch ν for the atom i, and the sum- mation is performed over all atoms within the unit cell. The...

    work page

  3. [2]

    I of the SM I)

    Two examples of the emergent particles enforced by Wyckoff position An example of subcase I in Extended Data Table 2: At the H point of the SG 220, three kinds of co-irreps can appear, which are {1, 2}, {3, 3}, and {4, 5} (Table 220 in Sec. I of the SM I). Among them, only co-irrep {4, 5} corresponds to SP; nevertheless, SP will all be present when atoms ...

    work page

  4. [3]

    (2) One one-dimensional (co- )irrep and one two-dimensional (co-)irrep

    The representations of the triply degenerate Goldstone modes The representations of the triply degenerate Goldstone modes at the Γ point may consist of: (1) One three- dimensional (co-)irrep. (2) One one-dimensional (co- )irrep and one two-dimensional (co-)irrep. (3) Three one- dimensional (co-)irreps. The representation of the acoustic triply point is on...

    work page

  5. [4]

    edu.cn/d/132f8ac8aac545f2862e/

    Supplemental Material See Supplemental Material at https://box.nju. edu.cn/d/132f8ac8aac545f2862e/. It is divided into three parts in total: Supplemental Material I, Supple- mental Material II, and Supplemental Material III. Supple- mental Material I contains the irreps along with the corre- sponding k · p models and the types of EMPs for each HSP in 230 ...

    work page 1943

  6. [5]

    Bansil, H

    A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys. 88, 021004 (2016)

    2016

  7. [6]

    X. Wan, A. M. Turner, A. Vishwanath, and S. Y . Savrasov, Topological semimetal and Fermi-arc surface states in the elec- tronic structure of pyrochlore iridates, Phys. Rev. B83, 205101 (2011)

    work page 2011

  8. [7]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insula- tors, Rev. Mod. Phys. 82, 3045 (2010)

    2010

Show all 77 references
  1. [8]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and supercon- ductors, Rev. Mod. Phys. 83, 1057 (2011)

    2011

  2. [9]

    T. O. Wehling, A. M. Black-Schaffer, and A. V . Balatsky, Dirac materials, Adv. Phys. 63, 1 (2014)

    2014

  3. [10]

    Y . Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. Vergniory, N. Regnault, Y . Chen, C. Felser, and B. A. Bernevig, High- throughput calculations of magnetic topological materials, Na- ture (London) 586, 702 (2020)

    2020

  4. [11]

    C.-K. Chiu, J. C. Y . Teo, A. P. Schnyder, and S. Ryu, Classifica- tion of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016)

    2016

  5. [12]

    N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018)

    2018

  6. [13]

    B. Q. Lv, T. Qian, and H. Ding, Experimental perspective on three-dimensional topological semimetals, Rev. Mod. Phys.93, 025002 (2021)

    2021

  7. [14]

    B. A. Bernevig, C. Felser, and H. Beidenkopf, Progress and prospects in magnetic topological materials, Nature (London) 603, 41 (2022)

    2022

  8. [15]

    Zhang, C.-X

    H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in Bi 2Se3, Bi 2Te3 and Sb 2Te3 with a single Dirac cone on the surface, Nat. phys. 5, 438 (2009)

    2009

  9. [16]

    M. G. Vergniory, B. J. Wieder, L. Elcoro, S. S. P. Parkin, C. Felser, B. A. Bernevig, and N. Regnault, All topological bands of all nonmagnetic stoichiometric materials, Science376, eabg9094 (2022)

    2022

  10. [17]

    F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Comprehensive search for topological materials using symmetry indicators, Na- ture (London) 566, 486 (2019)

    2019

  11. [18]

    M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high-quality topological materials, Nature (London) 566, 480 (2019)

    2019

  12. [19]

    Zhang, Y

    T. Zhang, Y . Jiang, Z. Song, H. Huang, Y . He, Z. Fang, H. Weng, and C. Fang, Catalogue of topological electronic ma- terials, Nature (London) 566, 475 (2019)

    2019

  13. [20]

    Z. Rao, H. Li, T. Zhang, S. Tian, C. Li, B. Fu, C. Tang, L. Wang, Z. Li, W. Fan, J. Li, Y . Huang, Z. Liu, Y . Long, C. Fang, H. Weng, Y . Shi, H. Lei, Y . Sun, T. Qian, and H. Ding, Obser- vation of unconventional chiral fermions with long Fermi arcs in CoSi, Nature (London) ...

    2019

  14. [21]

    Z. K. Liu, B. Zhou, Y . Zhang, Z. J. Wang, H. M. Weng, D. Prab- hakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y . L. Chen, Discovery of a Three-Dimensional Topological Dirac Semimetal, Na3Bi, Science 343, 864 (2014)

    2014

  15. [22]

    Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y . Zhang, H. M. Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hus- sain, and Y . L. Chen, A stable three-dimensional topological Dirac semimetal Cd3As2, Nat. Mater. 1...

    2014

  16. [23]

    B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Experimental Discovery of Weyl Semimetal TaAs, Phys. Rev. X5, 031013 (2015)

    2015

  17. [24]

    S.-Y . Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Dis- covery of a Weyl fermion semimetal and...

    2015

  18. [25]

    Z. Wang, Y . Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb), Phys. Rev. B 85, 195320 (2012)

    2012

  19. [26]

    J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, 2001)

    2001

  20. [27]

    Z. Zhu, G. W. Winkler, Q. S. Wu, J. Li, and A. A. Soluyanov, Triple Point Topological Metals, Phys. Rev. X 6, 031003 (2016)

    2016

  21. [28]

    H. Weng, C. Fang, Z. Fang, and X. Dai, Topological semimetals with triply degenerate nodal points in θ-phase tantalum nitride, Phys. Rev. B 93, 241202(R) (2016)

    2016

  22. [29]

    S. M. Young, S. Zaheer, J. C. Y . Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac Semimetal in Three Dimensions, Phys. Rev. Lett. 108, 140405 (2012)

    2012

  23. [30]

    B. J. Wieder, Y . Kim, A. M. Rappe, and C. L. Kane, Double Dirac Semimetals in Three Dimensions, Phys. Rev. Lett. 116, 186402 (2016)

    2016

  24. [31]

    Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Three- dimensional Dirac semimetal and quantum transport in Cd3As2, Phys. Rev. B 88, 125427 (2013)

    2013

  25. [32]

    C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry, Phys. Rev. Lett. 108, 266802 (2012)

    2012

  26. [33]

    J. Ahn, D. Kim, Y . Kim, and B.-J. Yang, Band Topology and Linking Structure of Nodal Line Semimetals withZ2 Monopole Charges, Phys. Rev. Lett. 121, 106403 (2018)

    2018

  27. [34]

    Bradlyn, J

    B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals, Science 353, aaf5037 (2016)

    2016

  28. [35]

    Y . Xu, M. G. Vergniory, D.-S. Ma, J. L. Ma ˜nes, Z.-D. Song, B. A. Bernevig, N. Regnault, and L. Elcoro, Catalog of topo- logical phonon materials, Science 384, eadf8458 (2024)

    2024

  29. [36]

    Watanabe, H

    H. Watanabe, H. C. Po, M. P. Zaletel, and A. Vishwanath, Filling-Enforced Gaplessness in Band Structures of the 230 Space Groups, Phys. Rev. Lett. 117, 096404 (2016)

    2016

  30. [37]

    R. Chen, H. C. Po, J. B. Neaton, and A. Vishwanath, Topolog- ical materials discovery using electron filling constraints, Nat. Phys. 14, 55 (2018)

    2018

  31. [38]

    Y . Liu, Y . Xu, and W. Duan, Berry phase and topological effects of phonons, Natl. Sci. Rev. 5, 314 (2018)

    2018

  32. [39]

    J. Li, J. Liu, S. A. Baronett, M. Liu, L. Wang, R. Li, Y . Chen, D. Li, Q. Zhu, and X.-Q. Chen, Computation and data driven discovery of topological phononic materials, Nat. Commun.12, 1204 (2021)

    2021

  33. [40]

    Singh, Q

    S. Singh, Q. S. Wu, C. Yue, A. H. Romero, and A. A. Soluyanov, Topological phonons and thermoelectricity in triple-point metals, Phys. Rev. Mater.2, 114204 (2018)

    2018

  34. [41]

    Zhang, Z

    T. Zhang, Z. Song, A. Alexandradinata, H. Weng, C. Fang, L. Lu, and Z. Fang, Double-Weyl Phonons in Transition-Metal Monosilicides, Phys. Rev. Lett. 120, 016401 (2018)

    2018

  35. [42]

    Q.-B. Liu, Z. Wang, and H.-H. Fu, Charge-four Weyl phonons, Phys. Rev. B 103, L161303 (2021)

    2021

  36. [43]

    Z. J. Chen, R. Wang, B. W. Xia, B. B. Zheng, Y . J. Jin, Y .- J. Zhao, and H. Xu, Three-Dimensional Dirac Phonons with 14 Inversion Symmetry, Phys. Rev. Lett.126, 185301 (2021)

    2021

  37. [44]

    C. Xie, Y . Liu, Z. Zhang, F. Zhou, T. Yang, M. Kuang, X. Wang, and G. Zhang, Sixfold degenerate nodal-point phonons: Sym- metry analysis and materials realization, Phys. Rev. B 104, 045148 (2021)

    2021

  38. [45]

    Zhang, J

    S. Zhang, J. Guan, X. Jia, B. Liu, W. Wang, F. Li, L. Wang, X. Ma, Q. Xue, J. Zhang, E. W. Plummer, X. Zhu, and J. Guo, Role of SrTiO3 phonon penetrating into thin FeSe films in the enhancement of superconductivity, Phys. Rev. B 94, 081116 (2016)

    2016

  39. [46]

    J. Zhu, W. Wu, J. Zhao, H. Chen, L. Zhang, and S. A. Yang, Symmetry-enforced nodal chain phonons, npj Quantum Mater. 7, 52 (2022)

    2022

  40. [47]

    Y . Liu, Y . Xu, S.-C. Zhang, and W. Duan, Model for topological phononics and phonon diode, Phys. Rev. B 96, 064106 (2017)

    2017

  41. [48]

    B. Li, J. Wang, L. Wang, and G. Zhang, Anomalous heat con- duction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels, Chaos 15, 015121 (2005)

    2005

  42. [49]

    Resta, S.-T

    G. Resta, S.-T. Pi, X. Wan, and S. Y . Savrasov, High surface conductivity of Fermi-arc electrons in Weyl semimetals, Phys. Rev. B 97, 085142 (2018)

    2018

  43. [50]

    J. Luo, T. Lin, J. Zhang, X. Chen, E. R. Blackert, R. Xu, B. I. Yakobson, and H. Zhu, Large effective magnetic fields from chiral phonons in rare-earth halides, Science 382, 698 (2023)

    2023

  44. [51]

    J. Li, H. Ma, Q. Xie, S. Feng, S. Ullah, R. Li, J. Dong, D. Li, Y . Li, and X.-Q. Chen, Topological quantum catalyst: Dirac nodal line states and a potential electrocatalyst of hydrogen evo- lution in the TiSi family, Sci. China Mater.61, 23 (2018)

    2018

  45. [52]

    Zhang and Q

    L. Zhang and Q. Niu, Angular Momentum of Phonons and the Einstein–de Haas Effect, Phys. Rev. Lett. 112, 085503 (2014)

    2014

  46. [53]

    H. Zhu, J. Yi, M.-Y . Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. Kaindl, L.-J. Li, Y . Wang, and X. Zhang, Observation of chiral phonons, Science 359, 579 (2018)

    2018

  47. [54]

    F. Wu, S. Bao, J. Zhou, Y . Wang, J. Sun, J. Wen, Y . Wan, and Q. Zhang, Fluctuation-enhanced phonon magnetic moments in a polar antiferromagnet, Nat. Phys. 19, 1868 (2023)

    2023

  48. [55]

    T. T. Zhang, H. Miao, Q. Wang, J. Q. Lin, Y . Cao, G. Fab- bris, A. H. Said, X. Liu, H. C. Lei, Z. Fang, H. M. Weng, and M. P. M. Dean, Phononic Helical Nodal Lines withPT Protec- tion in MoB2, Phys. Rev. Lett. 123, 245302 (2019)

    2019

  49. [56]

    Y . Gao, Y . Pan, J. Zhou, and L. Zhang, Chiral phonon mediated high-temperature superconductivity, Phys. Rev. B 108, 064510 (2023)

    2023

  50. [57]

    W. J. Choi, K. Yano, M. Cha, F. M. Colombari, J.-Y . Kim, Y . Wang, S. H. Lee, K. Sun, J. M. Kruger, A. F. de Moura, and N. A. Kotov, Chiral phonons in microcrystals and nanofibrils of biomolecules, Nat. Photonics 16, 366 (2022)

    2022

  51. [58]

    Fathizadeh, Phonon-assisted nearly pure spin current in DNA molecular chains: a multifractal analysis, Sci

    S. Fathizadeh, Phonon-assisted nearly pure spin current in DNA molecular chains: a multifractal analysis, Sci. Rep. 13, 21281 (2023)

    2023

  52. [59]

    H. Miao, T. T. Zhang, L. Wang, D. Meyers, A. H. Said, Y . L. Wang, Y . G. Shi, H. M. Weng, Z. Fang, and M. P. M. Dean, Observation of Double Weyl Phonons in Parity-Breaking FeSi, Phys. Rev. Lett. 121, 035302 (2018)

    2018

  53. [61]

    J. Li, J. Li, J. Tang, Z. Tao, S. Xue, J. Liu, H. Peng, X.-Q. Chen, J. Guo, and X. Zhu, Direct Observation of Topological Phonons in Graphene, Phys. Rev. Lett. 131, 116602 (2023)

    2023

  54. [62]

    Zhang, Z

    T. Zhang, Z. Huang, Z. Pan, L. Du, G. Zhang, and S. Murakami, Weyl Phonons in Chiral Crystals, Nano Lett. 23, 7561 (2023)

    2023

  55. [63]

    http://phonondb.mtl.kyoto-u.ac.jp/

  56. [64]

    Hellenbrandt, The inorganic crystal structure database (ICSD)—present and future, Crystallogr

    M. Hellenbrandt, The inorganic crystal structure database (ICSD)—present and future, Crystallogr. Rev. 10, 17 (2004)

    2004

  57. [65]

    Zhang and Q

    L. Zhang and Q. Niu, Chiral Phonons at High-Symmetry Points in Monolayer Hexagonal Lattices, Phys. Rev. Lett.115, 115502 (2015)

    2015

  58. [66]

    J. Zhou, L. Shen, M. D. Costa, K. A. Persson, S. P. Ong, P. Huck, Y . Lu, X. Ma, Y . Chen, H. Tang, and Y . P. Feng, 2DMatPedia, an open computational database of two-dimensional materials from top-down and bottom-up ap- proaches, Sci. Data 6, 86 (2019)

    2019

  59. [67]

    Jiang, U

    Y . Jiang, U. Petralanda, G. Skorupskii, Q. Xu, H. Pi, D. C˘alug˘aru, H. Hu, J. Xie, R. A. Mustaf, P. H ¨ohn, V . Haase, M. G. Vergniory, M. Claassen, L. Elcoro, N. Regnault, J. Shan, K. F. Mak, D. K. Efetov, E. Morosan, D. M. Kennes, A. Ru- bio, L. Xian, C. Felser, L. M. Scho...

    2024 arXiv

  60. [68]

    Petralanda, Y

    U. Petralanda, Y . Jiang, B. A. Bernevig, N. Regnault, and L. El- coro, Two-dimensional Topological Quantum Chemistry and Catalog of Topological Materials, arXiv:2411.08950 (2024)

    2024 arXiv

  61. [69]

    Zhang, J

    L. Zhang, J. Ren, J.-S. Wang, and B. Li, Topological Nature of the Phonon Hall Effect, Phys. Rev. Lett. 105, 225901 (2010)

    2010

  62. [70]

    Superconductor

    J. Goldstone, Field theories with “Superconductor” solutions, Il Nuovo Cimento (1955-1965) 19, 154 (1961)

    1961

  63. [71]

    X. Chen, X. Lu, S. Dubey, Q. Yao, S. Liu, X. Wang, Q. Xiong, L. Zhang, and A. Srivastava, Entanglement of single-photons and chiral phonons in atomically thin WSe2, Nat. Phys.15, 221 (2019)

    2019

  64. [72]

    Pan and F

    Y . Pan and F. Caruso, Vibrational dichroism of chiral valley phonons, Nano Lett. 23, 7463 (2023)

    2023

  65. [73]

    T. Wang, H. Sun, X. Li, and L. Zhang, Chiral phonons: Predic- tion, verification, and application, Nano Lett. 24, 4311 (2024)

    2024

  66. [74]

    Nambu and G

    Y . Nambu and G. Jona-Lasinio, Dynamical Model of Elemen- tary Particles Based on an Analogy with Superconductivity. I, Phys. Rev. 122, 345 (1961)

    1961

  67. [76]

    Goldstone, A

    J. Goldstone, A. Salam, and S. Weinberg, Broken Symmetries, Phys. Rev. 127, 965 (1962)

    1962

  68. [77]

    Togo and I

    A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015)

    2015

  69. [130]

    (b) The phonon dispersion of Zr 3N4 with MPID 11661 and SG 220

    The DP is marked by purple circle, the QDP is marked by red circle, the P-NS is marked by cyan circle, the P-NSs is marked by orange circle, the P-QNL is marked by blue circle and the P-WMLs is marked by green circle at the HSPs. (b) The phonon dispersion of Zr 3N4 with MPID 1...

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