Hourglass fermions


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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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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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.

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Correspondence to B. Andrei Bernevig.

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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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