Abstract

Chiral crystals are materials with a lattice structure that has a well-defined handedness due to the lack of inversion, mirror or other roto-inversion symmetries. Although it has been shown that the presence of crystalline symmetries can protect topological band crossings, the topological electronic properties of chiral crystals remain largely uncharacterized. Here we show that Kramers–Weyl fermions are a universal topological electronic property of all non-magnetic chiral crystals with spin–orbit coupling and are guaranteed by structural chirality, lattice translation and time-reversal symmetry. Unlike conventional Weyl fermions, they appear at time-reversal-invariant momenta. We identify representative chiral materials in 33 of the 65 chiral space groups in which Kramers–Weyl fermions are relevant to the low-energy physics. We determine that all point-like nodal degeneracies in non-magnetic chiral crystals with relevant spin–orbit coupling carry non-trivial Chern numbers. Kramers–Weyl materials can exhibit a monopole-like electron spin texture and topologically non-trivial bulk Fermi surfaces over an unusually large energy window.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Data availability

The data supporting the findings of this study are available within the paper and other findings of this study are available from the corresponding author upon reasonable request.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

  1. 1.

    Wigner, E. P. On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939).

  2. 2.

    Bradley, C. J. & Cracknell, A. P. The Mathematical Theory of Symmetry in Solids (Clarendon Press Oxford, Oxford, 1972).

  3. 3.

    Flack, H. D. Chiral and achiral crystal structure. Helv. Chim. Acta 86, 905–921 (2003).

  4. 4.

    Bogdanov, A. & Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Mater. 138, 255–269 (1994).

  5. 5.

    Rikken, G. L. J. A., Fölling, J. & Wyder, P. Electrical magnetochiral anisotropy. Phys. Rev. Lett. 87, 236602 (2001).

  6. 6.

    Yoda, T., Yokoyama, T. & Murakami, S. Current-induced orbital and spin magnetizations in crystals with helical structure. Sci. Rep. 5, 12024 (2015).

  7. 7.

    Fasman, G. D. Circular Dichroism and the Conformational Analysis of Biomolecules (Springer, Berlin, 2013).

  8. 8.

    Hasan, M. Z., Xu, S.-Y. & Bian, G. Topological insulators, topological superconductors and Weyl fermion semimetals. Phys. Scr. T164, 014001 (2015).

  9. 9.

    Zheng, H. & Hasan, M. Z. Quasiparticle interference on type-I and type-II Weyl semimetal surfaces: a review. Adv. Phys. X 3, 146661 (2018).

  10. 10.

    Yang, B. J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5, 4898 (2015).

  11. 11.

    Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356 (2007).

  12. 12.

    Wan, X. et al. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

  13. 13.

    Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).

  14. 14.

    Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).

  15. 15.

    Mañes, J. L. Existence of bulk chiral fermions and crystal symmetry. Phys. Rev. B 85, 155118 (2012).

  16. 16.

    Fang, C. et al. Multi-Weyl topological semimetals stabilized by point group symmetry. Phys. Rev. Lett. 107, 127205 (2011).

  17. 17.

    Kim, W., Wieder, B. J., Kane, C. L. & Rappe, A. M. Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 115, 036806 (2015).

  18. 18.

    Watanabe, H. et al. Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals. Proc. Natl Acad. Sci. USA 112, 14551–14556 (2015).

  19. 19.

    Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystal. Science 353, aaf5037 (2016).

  20. 20.

    Wieder, B. J. et al. Double Dirac semimetals in three dimensions. Phys. Rev. Lett. 116, 186402 (2016).

  21. 21.

    Po, H. C., Vishwanath, A. & Watanabe, H. Topological materials discovery using electron filling constraints. Nat. Phys. 14, 55–61 (2018).

  22. 22.

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

  23. 23.

    Chang, G. et al. Unconventional chiral fermions and large topological Fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).

  24. 24.

    Tang, P., Zhou, Q. & Zhang, S.-C. Multiple types of topological fermions in transition metal silicides. Phys. Rev. Lett. 119, 206402 (2017).

  25. 25.

    Witczak-Krempa, W., Knap, M. & Abanin, D. Interacting Weyl semimetals: characterization via the topological Hamiltonian and its breakdown. Phys. Rev. Lett. 113, 136402 (2014).

  26. 26.

    Bernevig, B. A. Lecture on Weyl semimetals at the Topological Matter School, Donostia International Physics Center (Topological Matter School, 2016); https://tms16.sciencesconf.org/; https://www.youtube.com/watch?v=j0zgWHLL1z4

  27. 27.

    Xiao, M. & Fan, S. Topologically charged nodal surface. Preprint at https://arxiv.org/abs/1709.02363 (2017).

  28. 28.

    Sharma, G. et al. Electronic structure, photovoltage, and photocatalytic hydrogen evolution with p-CuBi2O4 nanocrystals. J. Mater. Chem. A 4, 2936–2942 (2016).

  29. 29.

    Di Sante, D. et al. Realizing double Dirac particles in the presence of electronic interactions. Phys. Rev. B. 96, 121106(R) (2017).

  30. 30.

    Inorganic Crystal Structure Database (FIZ Karlsruhe, 2014); http://icsd.fiz-karlsruhe.de/icsd

  31. 31.

    Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

  32. 32.

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

  33. 33.

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

  34. 34.

    Kourtis, S. Bulk spectroscopic measurement of the topological charge of Weyl nodes with resonant x-rays. Phys. Rev. B 94, 125132 (2016).

  35. 35.

    Itoh, S. Weyl Fermions and spin dynamics of metallic ferromagnet SrRuO3. Nat. Commun. 7, 11788 (2016).

  36. 36.

    de Juan, F. et al. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Commun. 8, 15995 (2017).

  37. 37.

    Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2014).

  38. 38.

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

  39. 39.

    Chan, C.-K. & Lee, P. A. Emergence of bulk gap and metallic side walls in the zeroth Landau level in Dirac and Weyl semimetals. Phys. Rev. B 96, 195143 (2017).

  40. 40.

    Hu, J. et al. π Berry phase and Zeeman splitting of TaP probed by high field magnetotransport measurements. Sci. Rep. 6, 18674 (2016).

  41. 41.

    Wang, Z. & Zhang, S.-C. Chiral anomaly, charge density waves, and axion strings from Weyl semimetals. Phys. Rev. B 87, 161107(R) (2013).

  42. 42.

    Sun, Y., Zhang, Y., Felser, C. & Yan, B. Giant intrinsic spin Hall effect in the TaAs family of Weyl semimetals. Phys. Rev. Lett. 117, 146403 (2016).

  43. 43.

    Shan, J. & Heinz, T. F. Ultrafast Dynamical Processes in Semiconductors (Springer, Berlin, 2004).

  44. 44.

    Zyuzin, A. A. et al. Weyl semimetal with broken time reversal and inversion symmetries. Phys. Rev. B 85, 165110 (2012).

  45. 45.

    Zhong, S., Moore, J. E. & Souza, I. Gyrotropic magnetic effect and the magnetic moment on the Fermi surface. Phys. Rev. Lett. 116, 077201 (2016).

  46. 46.

    Bardarson, J. H., Lu, Y.-M. & Moore, J. E. Superconductivity of doped Weyl semimetals: finite-momentum pairing and electronic analogues of the 3He-A phase. Phys. Rev. B 86, 214514 (2012).

  47. 47.

    Hosur, P. & Qi, X.-L. Time-reversal invariant topological superconductivity in doped Weyl semimetals. Phys. Rev. B 90, 045130 (2014).

  48. 48.

    Xu, S.-Y. et al. Momentum-space imaging of Cooper pairing in a half-Dirac-gas topological superconductor. Nat. Phys. 10, 943–950 (2014).

  49. 49.

    Kresse, G. & Furthmöller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

  50. 50.

    Blaha, P., Schwarz, K. & Madsen, G. K. H. et al. An Augmented Plane Wave plus Local Orbital Program for Calculating Crystal Properties. (Vienna University of Technology, Vienna, 2001).

  51. 51.

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

Download references

Acknowledgements

Work at Princeton was supported by the US Department of Energy under Basic Energy Sciences (grant no. DOE/BES DE-FG-02-05ER46200). M.Z.H. acknowledges Visiting Scientist support from the Lawrence Berkeley National Laboratory, and partial support for theoretical work from the Gordon and Betty Moore Foundation (grant no. GBMF4547/Hasan). The work at the National University of Singapore was supported by the National Research Foundation, Prime Minister’s Office, Singapore, under its NRF fellowship (NRF award no. NRF-NRFF2013-03). B.J.W. acknowledges support through a Simons Investigator grant from the Simons Foundation to C. L. Kane, through Nordita under ERC DM 321031, through grants from the Department of Energy (no. DE-SC0016239), the Simons Foundation (Simons Investigator grant no. ONR-N00014-14-1-0330), the Packard Foundation and the Schmidt Fund to B. A. Bernevig, and acknowledges the hospitality of the Donostia International Physics Center. F.S. and T.N. acknowledge support by the Swiss National Science Foundation (grant no. 200021–169061) and the ERC-StG-Neupert-757867-PARATOP, respectively. T.-R.C. was supported by the Ministry of Science and Technology under the MOST Young Scholar Fellowship: MOST Grant for the Columbus Program no. 107-2636-M-006-004-, National Cheng Kung University, Taiwan, and the National Center for Theoretical Sciences (NCTS), Taiwan. M.Z.H. acknowledges support from the Miller Institute of Basic Research in Science at the University of California at Berkeley in the form of a Visiting Miller Professorship during the early stages of this work. The authors thank C. L. Kane and R. Kamien for helpful discussions on chirality and thank B. Bradlyn, J. Cano, M. I. Aroyo and B. A. Bernevig for insightful discussions on group theory and symmetry.

Author information

Author notes

  1. These authors contributed equally: Guoqing Chang, Benjamin J. Wieder, Frank Schindler.

Affiliations

  1. Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, NJ, USA

    • Guoqing Chang
    • , Daniel S. Sanchez
    • , Ilya Belopolski
    • , Su-Yang Xu
    •  & M. Zahid Hasan
  2. Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, Singapore, Singapore

    • Guoqing Chang
    • , Bahadur Singh
    • , Di Wu
    •  & Hsin Lin
  3. Department of Physics, National University of Singapore, Singapore, Singapore

    • Guoqing Chang
    • , Bahadur Singh
    • , Di Wu
    •  & Hsin Lin
  4. Institute of Physics, Academia Sinica, Taipei, Taiwan

    • Guoqing Chang
    •  & Hsin Lin
  5. Department of Physics, Princeton University, Princeton, NJ, USA

    • Benjamin J. Wieder
  6. Nordita, Center for Quantum Materials, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden

    • Benjamin J. Wieder
  7. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA

    • Benjamin J. Wieder
  8. Department of Physics, University of Zurich, Zurich, Switzerland

    • Frank Schindler
    •  & Titus Neupert
  9. Department of Physics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    • Shin-Ming Huang
  10. Department of Physics, National Cheng Kung University, Tainan, Taiwan

    • Tay-Rong Chang
  11. Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    • M. Zahid Hasan

Authors

  1. Search for Guoqing Chang in:

  2. Search for Benjamin J. Wieder in:

  3. Search for Frank Schindler in:

  4. Search for Daniel S. Sanchez in:

  5. Search for Ilya Belopolski in:

  6. Search for Shin-Ming Huang in:

  7. Search for Bahadur Singh in:

  8. Search for Di Wu in:

  9. Search for Tay-Rong Chang in:

  10. Search for Titus Neupert in:

  11. Search for Su-Yang Xu in:

  12. Search for Hsin Lin in:

  13. Search for M. Zahid Hasan in:

Contributions

All the authors contributed to the intellectual content of this work. By systematically studying the electronic structures of chiral crystals, the existence of Weyl points at TRIM points of chiral crystals was recognized by G.C. and S.-Y.X. in consultation with M.Z.H. B.J.W. proved that Weyl fermions at TRIM points (Kramers-Weyl fermions) are a generic feature of all chiral crystals, and thus that all point degeneracies in chiral crystals are topological. F.S. and T.N. proved the relationship between bulk symmetry eigenvalues and the chiral charge of Kramers–Weyl fermions. Spin–momentum locking was proposed by F.S. and T.N., and applied to models and materials by B.J.W., S.-Y.X and G.C. The materials search was done by G.C. and S.-Y.X. with help from all the authors. Tight-binding models were constructed by B.J.W., F.S. and T.N. The first-principles calculations were performed by G.C., S.-M.H., B.S., D.W.,T.-R.C. and H.L. The manuscript was written by G.C., B.J.W., F.S., T.N., S.-Y.X., H.L. and M.Z.H. with the help of D.S.S. and I.B. S.-Y.X., H.L. and M.Z.H. were responsible for the overall research direction, planning and integration among different research units.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Su-Yang Xu or Hsin Lin or M. Zahid Hasan.

Supplementary information

  1. Supplementary Information

    Supplementary Text, Supplementary Tables 1–4, Supplementary Figures 1–12 and Supplementary References 1–32

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/s41563-018-0169-3