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Hourglass fermions

Nature volume 532, pages 189–194 (2016)Cite this article

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Abstract

Spatial symmetries in crystals may be distinguished by whether they preserve the spatial origin. Here we study spatial symmetries that translate the origin by a fraction of the lattice period, and find that these non-symmorphic symmetries protect an exotic surface fermion whose dispersion relation is shaped like an hourglass; surface bands connect one hourglass to the next in an unbreakable zigzag pattern. These ‘hourglass’ fermions are formed in the large-gap insulators, KHgX (X = As, Sb, Bi), which we propose as the first material class whose band topology relies on non-symmorphic symmetries. Besides the hourglass fermion, another surface of KHgX manifests a three-dimensional generalization of the quantum spin Hall effect, which has previously been observed only in two-dimensional crystals. To describe the bulk topology of non-symmorphic crystals, we propose a non-Abelian generalization of the geometric theory of polarization. Our non-trivial topology originates from an inversion of the rotational quantum numbers, which we propose as a criterion in the search for topological materials.

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Figure 1: Hourglass fermions and the 3D quantum spin Hall effect (QSHE).
Figure 2: Crystal structure and Brillouin zone of KHgX.
Figure 3: The 010-surface band structure.
Figure 4: The 100-surface band structure.
Figure 5: Bulk band structure and orbital analysis.
Figure 6: Spectra of the projected-position operator.

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Acknowledgements

We thank C. Fang, D. P. Arovas, J. Li, L. Muechler and X. Dai for discussions. This work was supported by NSF CAREER DMR-095242, ONR-N00014-11-1-0635, TI MURI W911NF-12-1-0461, NSF-MRSEC DMR-1420541, Packard Foundation, Keck grant, “ONR Majorana Fermions” 25812-G0001-10006242-101, and Schmidt fund 23800-E2359-FB625. During the refereeing stages of this work, A.A. was supported by the Yale Fellowship in Condensed Matter Physics.

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Author notes

  1. Zhijun Wang and A. Alexandradinata: These authors contributed equally to this work.

Authors and Affiliations

  1. Department of Physics, Princeton University, Princeton, 08544, New Jersey, USA

    Zhijun Wang, A. Alexandradinata & B. Andrei Bernevig

  2. Department of Physics, Yale University, New Haven, 06520, Connecticut, USA

    A. Alexandradinata

  3. Department of Chemistry, Princeton University, Princeton, 08540, New Jersey, USA

    R. J. Cava

Authors
  1. Zhijun Wang
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  2. A. Alexandradinata
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  4. B. Andrei Bernevig
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Contributions

A.A., Z.W. and B.A.B. performed theoretical analysis; Z.W. discovered the KHgX material class and performed the first-principles calculations; R.J.C. provided several other material suggestions.

Corresponding author

Correspondence to B. Andrei Bernevig.

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The authors declare no competing financial interests.

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This file contains Supplementary Text and Data 1-5, Supplementary Figures 1-9, Supplementary Tables 1-3 and additional references. (PDF 6069 kb)

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Wang, Z., Alexandradinata, A., Cava, R. et al. Hourglass fermions. Nature 532, 189–194 (2016). https://doi.org/10.1038/nature17410

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