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# Topological quantum properties of chiral crystals

## 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.

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## 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.

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## 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

Authors

### 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.

### Corresponding authors

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

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

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

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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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