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Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe

Nature Physics volume 16, pages 1137–1143 (2020)Cite this article

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Abstract

Weyl semimetals in three-dimensional crystals provide the paradigm example of topologically protected band nodes. It is usually taken for granted that a pair of colliding Weyl points annihilate whenever they carry opposite chiral charge. In stark contrast, here we report that Weyl points in systems that are symmetric under the composition of time reversal with a π rotation are characterized by a non-Abelian topological invariant. The topological charges of the Weyl points are transformed via braid phase factors, which arise upon exchange inside symmetric planes of the reciprocal momentum space. We elucidate this process with an elementary two-dimensional tight-binding model that is implementable in cold-atom set-ups and in photonic systems. In three dimensions, interplay of the non-Abelian topology with point-group symmetry is shown to enable topological phase transitions in which pairs of Weyl points may scatter or convert into nodal-line rings. By combining our theoretical arguments with first-principles calculations, we predict that Weyl points occurring near the Fermi level of zirconium telluride carry non-trivial values of the non-Abelian charge, and that uniaxial compression strain drives a non-trivial conversion of the Weyl points into nodal lines.

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Fig. 1: Reciprocal braiding of band nodes.
Fig. 2: Elementary protocol for braiding band nodes.
Fig. 3: Frame-rotation charge.
Fig. 4: Berry curvature versus Euler form.
Fig. 5: Conversion of Weyl points in 3D momentum space.
Fig. 6: Weyl points and nodal lines in ZrTe.

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Data availability

All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request. Input files used to generate the first-principles data are provided in the Materials Cloud Archive71.

Code availability

The Mathematica notebook for computing the Euler class of a collection of band nodes by implementing the method presented in Supplementary Section H is available online65.

References

  1. Wen, X.-G. Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405–473 (1995).

    ADS  Google Scholar 

  2. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Google Scholar 

  3. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    ADS  Google Scholar 

  4. Fu, L. & Kane, C. L. Superconducting proximity effect and majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).

    ADS  Google Scholar 

  5. Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).

    ADS  MATH  Google Scholar 

  6. Ryu, S., Schnyder, A., Furusaki, A. & Ludwig, A. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).

    ADS  Google Scholar 

  7. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).

    ADS  Google Scholar 

  8. Slager, R.-J., Mesaros, A., Juričić, V. & Zaanen, J. The space group classification of topological band-insulators. Nat. Phys. 9, 98–102 (2012).

    Google Scholar 

  9. Bzdušek, T., Wu, Q.-S., Rüegg, A., Sigrist, M. & Soluyanov, A. A. Nodal-chain metals. Nature 538, 75–78 (2016).

    ADS  Google Scholar 

  10. Fang, C., Weng, H., Dai, X. & Fang, Z. Topological nodal line semimetals. Chin. Phys. B 25, 117106 (2016).

    ADS  Google Scholar 

  11. Bzdušek, T. & Sigrist, M. Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems. Phys. Rev. B 96, 155105 (2017).

    ADS  Google Scholar 

  12. Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. L. & Slager, R.-J. Topological classification of crystalline insulators through band structure combinatorics. Phys. Rev. X 7, 041069 (2017).

    Google Scholar 

  13. Bouhon, A. & Black-Schaffer, A. M. Global band topology of simple and double Dirac-point semimetals. Phys. Rev. B 95, 241101 (2017).

    ADS  Google Scholar 

  14. Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).

    ADS  Google Scholar 

  15. Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    ADS  Google Scholar 

  16. Slager, R.-J. The translational side of topological band insulators. J. Phys. Chem. Solids 128, 24–38 (2019).

    ADS  Google Scholar 

  17. Höller, J. & Alexandradinata, A. Topological Bloch oscillations. Phys. Rev. B 98, 024310 (2018).

    ADS  Google Scholar 

  18. Zhang, T. et al. Catalogue of topological electronic materials. Nature 566, 475–479 (2019).

    ADS  Google Scholar 

  19. Autes, G., Wu, Q., Mounet, N. & Yazyev, O. V. TopoMat: a database of high-throughput first-principles calculations of topological materials. Materials Cloud Archive 2019.0019/v2 (2019); https://doi.org/10.24435/materialscloud:2019.0019/v2

  20. Wan, X., Turner, A. M., Vishwanath, A. & Sergey Y., S. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    ADS  Google Scholar 

  21. Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).

    Google Scholar 

  22. Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).

    Google Scholar 

  23. Su-YangXu, I. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

    ADS  Google Scholar 

  24. Huang, X. et al. Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).

    Google Scholar 

  25. Poenaru, V. & Toulouse, G. The crossing of defects in ordered media and the topology of 3-manifolds. J. Phys. Lett. 38, 887–895 (1977).

    MathSciNet  Google Scholar 

  26. Volovik, G. E. & Mineev, V. P. Investigation of singularities in superfluid He3 in liquid crystals by the homotopic topology methods. Zh. Eksp. Teor. Fiz. 72, 2256–2274 (1977).

    Google Scholar 

  27. Madsen, L. A., Dingemans, T. J., Nakata, M. & Samulski, E. T. Thermotropic biaxial nematic liquid crystals. Phys. Rev. Lett. 92, 145505 (2004).

    ADS  Google Scholar 

  28. Alexander, G. P., Chen, B. G.-G., Matsumoto, E. A. & Kamien, R. D. Colloquium: disclination loops, point defects and all that in nematic liquid crystals. Rev. Mod. Phys. 84, 497–514 (2012).

    ADS  Google Scholar 

  29. Liu, K., Nissinen, J., Slager, R.-J., Wu, K. & Zaanen, J. Generalized liquid crystals: giant fluctuations and the vestigial chiral order of I, O and T matter. Phys. Rev. X 6, 041025 (2016).

    Google Scholar 

  30. Wu, Q., Soluyanov, A. A. & Bzdušek, T. Non-Abelian band topology in noninteracting metals. Science 365, 1273–1277 (2019).

    MathSciNet  MATH  Google Scholar 

  31. Tiwari, A. & Bzdušek, T. Non-Abelian topology of nodal-line rings in \({\mathcal{PT}}\)-symmetric systems. Phys. Rev. B 101, 195130 (2020).

    ADS  Google Scholar 

  32. Ahn, J., Park, S. & Yang, B.-J. Failure of Nielsen–Ninomiya theorem and fragile topology in two-dimensional systems with space–time inversion symmetry: application to twisted bilayer graphene at magic angle. Phys. Rev. X 9, 021013 (2019).

    Google Scholar 

  33. Ahn, J., Kim, D., Youngkuk, K. & Yang, B.-J. Band topology and linking structure of nodal line semimetals with Z2 monopole charges. Phys. Rev. Lett. 121, 106403 (2018).

    ADS  Google Scholar 

  34. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    ADS  Google Scholar 

  35. Po, H. C., Zou, L., Senthil, T. & Vishwanath, A. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).

    ADS  Google Scholar 

  36. Song, Z., Wang, Z., Shi, W., Li, G., Fang, C. & Bernevig, B. A. All magic angles in twisted bilayer graphene are topological. Phys. Rev. Lett. 123, 036401 (2019).

    ADS  Google Scholar 

  37. Bouhon, A., Black-Schaffer, A. M. & Slager, R.-J. Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry. Phys. Rev. B 100, 195135 (2019).

    ADS  Google Scholar 

  38. Francis, G. K. & Kauffman, L. H. Air on the Dirac strings. Contemp. Math. 169, 261–276 (1994).

    MathSciNet  MATH  Google Scholar 

  39. Johansson, N. & Sjöqvist, E. Optimal topological test for degeneracies of real Hamiltonians. Phys. Rev. Lett. 92, 060406 (2004).

    ADS  Google Scholar 

  40. Zhao, Y. X. & Lu, Y. PT-symmetric real Dirac fermions and semimetals. Phys. Rev. Lett. 118, 056401 (2017).

    ADS  Google Scholar 

  41. Milnor, J. W. & Stasheff, J. D. Characteristic Classes (Princeton Univ. Press, 1974).

  42. Fukui, T., Hatsugai, Y. & Suzuki, H. Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances. J. Phys. Soc. Jpn 74, 1674–1677 (2005).

    ADS  Google Scholar 

  43. Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015).

    ADS  Google Scholar 

  44. Lv, B. Q. et al. Observation of three-component fermions in the topological semimetal molybdenum phosphide. Nature 546, 627–631 (2017).

    ADS  Google Scholar 

  45. Sun, X.-Q., Zhang, S.-C. & Bzdušek, T. Conversion rules for Weyl points and nodal lines in topological media. Phys. Rev. Lett. 121, 106402 (2018).

    ADS  Google Scholar 

  46. Zhu, Z., Winkler, G. W., Wu, Q., Li, J. & Soluyanov, A. A. Triple point topological metals. Phys. Rev. X 6, 031003 (2016).

    Google Scholar 

  47. Weng, H., Fang, C., Fang, Z. & Dai, X. Coexistence of Weyl fermion and massless triply degenerate nodal points. Phys. Rev. B 94, 165201 (2016).

    ADS  Google Scholar 

  48. He, J. B. et al. Magnetotransport properties of the triply degenerate node topological semimetal tungsten carbide. Phys. Rev. B 95, 195165 (2017).

    ADS  Google Scholar 

  49. Ma, J.-Z. et al. Three-component fermions with surface Fermi arcs in tungsten carbide. Nat. Phys. 14, 349–354 (2018).

    Google Scholar 

  50. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    ADS  Google Scholar 

  51. Mostofi, A. A. et al. An updated version of Wannier90: a tool for obtaining maximally-localised Wannier functions. Comp. Phys. Commun. 185, 2309–2310 (2014).

    ADS  MATH  Google Scholar 

  52. Wu, Q., Zhang, S., Song, H.-F., Troyer, M. & Soluyanov, A. A. WannierTools: an open-source software package for novel topological materials. Comp. Phys. Commun. 224, 405–416 (2018).

    ADS  Google Scholar 

  53. Guo, S.-D., Wang, Y.-H. & Lu, W.-L. Elastic and transport properties of topological semimetal ZrTe. N. J. Phys. 19, 113044 (2017).

    Google Scholar 

  54. Sie, E. J. et al. An ultrafast symmetry switch in a Weyl semimetal. Nature 565, 61–66 (2019).

    ADS  Google Scholar 

  55. Li, T. et al. Bloch state tomography using Wilson lines. Science 352, 1094–1097 (2016).

    ADS  Google Scholar 

  56. Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984).

    ADS  MathSciNet  Google Scholar 

  57. Fläschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).

    ADS  Google Scholar 

  58. Lu, L., Fu, L., Joannopoulos, J. D. & Soljačič, M. Weyl points and line nodes in gyroid photonic crystals. Nat. Photon. 7, 294–299 (2013).

    ADS  Google Scholar 

  59. Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  60. Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59–62 (2018).

    ADS  Google Scholar 

  61. Nakahara, M. Geometry, Topology and Physics (Taylor & Francis, 2003).

  62. Hatcher, A. Vector Bundles and K-Theory (Cornell Univ., 2017); https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf

  63. Chern, S.-S. On the curvatura integra in a Riemannian manifold. Ann. Math. 46, 674–684 (1945).

    MathSciNet  MATH  Google Scholar 

  64. Hatcher, A. Algebraic Topology (Cambridge Univ. Press, 2002).

  65. Bzdušek, T. Euler Class of a Pair of Energy Bands on a Manifold with a Boundary (ResearchGate, 2019); https://doi.org/10.13140/RG.2.2.29803.69928

  66. Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).

    ADS  Google Scholar 

  67. Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

    ADS  Google Scholar 

  68. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    ADS  Google Scholar 

  69. Krukau, A. V., Vydrov, O. A., Izmaylov, A. F. & Scuseria, G. E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys. 125, 224106 (2006).

    ADS  Google Scholar 

  70. Örlygsson, G. & Harbrecht, B. The crystal structure of WC type ZrTe. Advantages in chemical bonding as contrasted to NiAs type ZrTe. Z. Naturforsch. B 54, 1125–1128 (1999).

    Google Scholar 

  71. Bouhon, A. et al. Supplementary data for ‘non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe’. Materials Cloud Archive 2020.54 (2020); https://doi.org/10.24435/materialscloud:vb-mk

Download references

Acknowledgements

We acknowledge valuable discussions with C.C. Wojcik, A. Vishwanath and B.A. Bernevig. R.-J.S. acknowledges funding via A. Vishwanath from the Center for Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the US Department of Energy Office of Science, Office of Basic Energy Sciences, through the Ames Laboratory under contract no. DE-AC02-07CH11358, the Marie Skłodowska-Curie programme under EC grant no. 842901, and Trinity College as well as the Winton programme at the University of Cambridge. T.B. was supported by the Gordon and Berry Moore Foundation’s EPiQS Initiative, grant no. GBMF4302, and by the Ambizione Program of the Swiss National Science Foundation, grant no. 185806. Q.-S.W. and O.V.Y. acknowledge support from NCCR Marvel. H.M.W. acknowledges support from the Ministry of Science and Technology of China under grants 2018YFA0305700 and 2016YFA0300600, the Strategic Priority Research Program of Chinese Academy of Sciences (grant no. XDB33000000) and the K. C. Wong Education Foundation (GJTD-2018-01). First-principles calculations have been performed at the Swiss National Supercomputing Centre (CSCS) under project no. s832 and s1008 and the facilities of Scientific IT and Application Support Center of EPFL.

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Author notes

  1. These authors contributed equally: Adrien Bouhon, QuanSheng Wu, Robert-Jan Slager.

Authors and Affiliations

  1. Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden

    Adrien Bouhon

  2. Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden

    Adrien Bouhon

  3. Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    QuanSheng Wu & Oleg V. Yazyev

  4. National Centre for Computational Design and Discovery of Novel Materials MARVEL, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    QuanSheng Wu & Oleg V. Yazyev

  5. TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Robert-Jan Slager

  6. Department of Physics, Harvard University, Cambridge, MA, UK

    Robert-Jan Slager

  7. Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Hongming Weng

  8. Songshan Lake Materials Laboratory, Guangdong, China

    Hongming Weng

  9. Condensed Matter Theory Group, Paul Scherrer Institute, Villigen, Switzerland

    Tomáš Bzdušek

  10. Department of Physics, University of Zürich, Zürich, Switzerland

    Tomáš Bzdušek

  11. Department of Physics, McCullough Building, Stanford University, Stanford, CA, USA

    Tomáš Bzdušek

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  1. Adrien Bouhon
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Contributions

A.B., R.-J.S. and T.B. contributed equally to the theoretical analysis in this work and wrote the manuscript. Q.-S.W. discovered the nodal conversion in ZrTe and obtained the presented first-principles data. H.M.W. and O.V.Y. were involved in the discussion and analysis of the first-principles data. All authors discussed and commented on the manuscript.

Corresponding authors

Correspondence to Adrien Bouhon, QuanSheng Wu or Robert-Jan Slager.

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Extended data

Extended Data Fig. 1 Wilson loop computation of Euler class of two principal nodes in the presence of two adjacent nodes.

a-b Two possible flows of a base loop (lν, ν \(\in\) [0, 1]) over the punctured Brillouin zone containing two principal nodes (black) and excluding two adjacent nodes (grey). The flow starts at a base point (l0 = x0) and ends at the boundary \({l}_{1}=\partial {\rm{BZ}}-\partial {{\mathcal{D}}}^{\epsilon }\), with the orange arrow indicating the direction of the flow. Assuming that the pair of principal nodes were created first (along the dashed line between them), the Euler class is \(\chi [{\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }]=0\) in case a, and \(\chi [{\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }]=1\) in case b. c-f Wilson loop (c,d) and Pfaffian (e,f) as a function of the flow of base loop over the punctured Brillouin zone, that is corresponding to a for (c,e), and to b for (d,f). e The zero winding of the Pfaffian, Δζ = 0, indicates a trivial frame-rotation charge of the two principal nodes (that is zero Euler class on \({\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }\)). f The non-zero winding of the Pfaffian, Δζ/(2π) = 1, indicates a non-trivial frame-rotation charge of the two principal nodes (that is non-vanishing Euler class on \({\rm{BZ}}-{{\mathcal{D}}}^{\epsilon }\)).

Extended Data Fig. 2 Fermi surface and band structures of ZrTe with and without strain.

a. Fermi surface of ZrTe under ambient conditions, with pairs of Weyl nodes of opposite chirality located inside the cyan pockets close to kz = 0 plane. b and c. Band structures of ZrTe along high-symmetry lines of the Brillouin under 0% resp. under 2.6% uniaxial compression strain. Path M-K-Γ lies within the kz = 0 mirror-invariant plane. The mirror eigenvalues + i and − i are indicated by the red vs. blue colour of the corresponding energy band. The plotted data were obtained with PBE+HSE06 functional.

Extended Data Fig. 3 Band structure of ZrTe.

Band structure of ZrTe as computed from PBE+HSE06 (left) resp. from PBE (right) functional.

Extended Data Fig. 4 Comparison of principal band nodes of ZrTe and MoP.

Band structure and location of nodal points of ZrTe and MoP, obtained with PBE+HSE06 functional. The band nodes in panels c and d correspond to degeneracies of energy bands marked with green and red colour in panels a and b.

Extended Data Fig. 5 Band structure and principal band nodes of ambient TaAs and with strain.

Band structure and band node locations in the first Brillouin zone of TaAs a–d under ambient conditions, and e–h with 5% [001] uniaxial strain obtained with PBE functional. The first column shows the band structure along the ΓX line. The next three columns show the location of band nodes in 3-dimension view, top view, and front view, respectively. Weyl points with chirality +1 and -1 are respectively indicated with magenta and green spheres. Nodal lines of the strained TaAs are indicated as brown lines. The \({C}_{2}{\mathcal{T}}\)-invariant plane is indicated with cyan colour, while the vertical mirror planes are displayed in shades of yellow.

Extended Data Fig. 6 Euler curvature in TaAs.

Euler curvature of a pair of Weyl points inside the \({C}_{2}{\mathcal{T}}\)-invariant plane of TaAs under ambient conditions. For a discussion, see Methods.

Extended Data Fig. 7 Band structure of nine WC-type materials obtained with PBE functional.

The corresponding nodal structure near the K-point under ambient conditions is indicated in each panel.

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Bouhon, A., Wu, Q., Slager, RJ. et al. Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe. Nat. Phys. 16, 1137–1143 (2020). https://doi.org/10.1038/s41567-020-0967-9

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