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.
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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.
Wigner, E. P. On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939).
Bradley, C. J. & Cracknell, A. P. The Mathematical Theory of Symmetry in Solids (Clarendon Press Oxford, Oxford, 1972).
Flack, H. D. Chiral and achiral crystal structure. Helv. Chim. Acta 86, 905–921 (2003).
Bogdanov, A. & Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Mater. 138, 255–269 (1994).
Rikken, G. L. J. A., Fölling, J. & Wyder, P. Electrical magnetochiral anisotropy. Phys. Rev. Lett. 87, 236602 (2001).
Yoda, T., Yokoyama, T. & Murakami, S. Current-induced orbital and spin magnetizations in crystals with helical structure. Sci. Rep. 5, 12024 (2015).
Fasman, G. D. Circular Dichroism and the Conformational Analysis of Biomolecules (Springer, Berlin, 2013).
Hasan, M. Z., Xu, S.-Y. & Bian, G. Topological insulators, topological superconductors and Weyl fermion semimetals. Phys. Scr. T164, 014001 (2015).
Zheng, H. & Hasan, M. Z. Quasiparticle interference on type-I and type-II Weyl semimetal surfaces: a review. Adv. Phys. X 3, 146661 (2018).
Yang, B. J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5, 4898 (2015).
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).
Wan, X. et al. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).
Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).
Mañes, J. L. Existence of bulk chiral fermions and crystal symmetry. Phys. Rev. B 85, 155118 (2012).
Fang, C. et al. Multi-Weyl topological semimetals stabilized by point group symmetry. Phys. Rev. Lett. 107, 127205 (2011).
Kim, W., Wieder, B. J., Kane, C. L. & Rappe, A. M. Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 115, 036806 (2015).
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).
Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystal. Science 353, aaf5037 (2016).
Wieder, B. J. et al. Double Dirac semimetals in three dimensions. Phys. Rev. Lett. 116, 186402 (2016).
Po, H. C., Vishwanath, A. & Watanabe, H. Topological materials discovery using electron filling constraints. Nat. Phys. 14, 55–61 (2018).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).
Chang, G. et al. Unconventional chiral fermions and large topological Fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).
Tang, P., Zhou, Q. & Zhang, S.-C. Multiple types of topological fermions in transition metal silicides. Phys. Rev. Lett. 119, 206402 (2017).
Witczak-Krempa, W., Knap, M. & Abanin, D. Interacting Weyl semimetals: characterization via the topological Hamiltonian and its breakdown. Phys. Rev. Lett. 113, 136402 (2014).
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
Xiao, M. & Fan, S. Topologically charged nodal surface. Preprint at https://arxiv.org/abs/1709.02363 (2017).
Sharma, G. et al. Electronic structure, photovoltage, and photocatalytic hydrogen evolution with p-CuBi2O4 nanocrystals. J. Mater. Chem. A 4, 2936–2942 (2016).
Di Sante, D. et al. Realizing double Dirac particles in the presence of electronic interactions. Phys. Rev. B. 96, 121106(R) (2017).
Inorganic Crystal Structure Database (FIZ Karlsruhe, 2014); http://icsd.fiz-karlsruhe.de/icsd
Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).
Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
Kourtis, S. Bulk spectroscopic measurement of the topological charge of Weyl nodes with resonant x-rays. Phys. Rev. B 94, 125132 (2016).
Itoh, S. Weyl Fermions and spin dynamics of metallic ferromagnet SrRuO3. Nat. Commun. 7, 11788 (2016).
de Juan, F. et al. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Commun. 8, 15995 (2017).
Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2014).
Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015).
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).
Hu, J. et al. π Berry phase and Zeeman splitting of TaP probed by high field magnetotransport measurements. Sci. Rep. 6, 18674 (2016).
Wang, Z. & Zhang, S.-C. Chiral anomaly, charge density waves, and axion strings from Weyl semimetals. Phys. Rev. B 87, 161107(R) (2013).
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).
Shan, J. & Heinz, T. F. Ultrafast Dynamical Processes in Semiconductors (Springer, Berlin, 2004).
Zyuzin, A. A. et al. Weyl semimetal with broken time reversal and inversion symmetries. Phys. Rev. B 85, 165110 (2012).
Zhong, S., Moore, J. E. & Souza, I. Gyrotropic magnetic effect and the magnetic moment on the Fermi surface. Phys. Rev. Lett. 116, 077201 (2016).
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).
Hosur, P. & Qi, X.-L. Time-reversal invariant topological superconductivity in doped Weyl semimetals. Phys. Rev. B 90, 045130 (2014).
Xu, S.-Y. et al. Momentum-space imaging of Cooper pairing in a half-Dirac-gas topological superconductor. Nat. Phys. 10, 943–950 (2014).
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).
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).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
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.
The authors declare no competing interests.
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Chang, G., Wieder, B.J., Schindler, F. et al. Topological quantum properties of chiral crystals. Nature Mater 17, 978–985 (2018). https://doi.org/10.1038/s41563-018-0169-3
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