Research ArticleQuantum Mechanics

Filling-enforced quantum band insulators in spin-orbit coupled crystals

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Science Advances  08 Apr 2016:
Vol. 2, no. 4, e1501782
DOI: 10.1126/sciadv.1501782
  • Fig. 1 Filling-enforced quantum band insulators.

    (A to C) Schematic band diagrams. Starting from a “primitive motif,” one can generate a crystal by repeating it according to the SG symmetries. Depending on the nature of the material, the motif can correspond to atoms, molecules, or bonding orbitals. Band theory accounts for how the electronic levels evolve when the number of motifs N is varied from 1 to ∞, during which the single-particle electronic wave function Embedded Image evolves from localized to delocalized. (A) In a strict AI, the set of filled bands, as indicated by the Fermi energy EF, has a natural correspondence to the filled electronic level of a single motif and therefore admits an equally valid classical, localized representation. (B) For a general band insulator (including all previously known quantum band insulators), a set of electronic levels are fully filled in the single-motif limit, although the filled bands do not have a simple correspondence with them. (C) feQBIs are crystals in which no symmetry-respecting choice of motif will have fully filled local energy levels, but nonetheless, a band insulator is made possible by quantum mechanical interference of the electron waves. (D) Nearest-neighbor hyperkagome lattice considered in Eq. 1 for SG No. 199. Blue spheres correspond to sites assigned to the displayed conventional cubic unit cell, which has twice the volume of a primitive unit cell. Transparent spheres represent sites in adjacent cells. (E and F) Example band structures for the tight-binding model in Eq. 1 with filling ν = 4. The lattice constant is set to 1. With spin-orbit coupling in (E), the system is insulating and forms an feQBI. When spin-orbit coupling is switched off in (F), the lowest four bands are completely flat and touch the upper bands at Embedded Image, rendering the system semimetallic.

  • Fig. 2 SE cut and entanglement spectrum.

    (A) Schematic of the SE cut in which the occupancy of one spin species (say ⇓) is traced over. (B) Example SE cut spectrum for the feQBI model of SG No. 199. As detailed in the Supplementary Materials, the entanglement band structure inherits little group representations from the physical bands, and TR symmetry is realized in a “particle-hole” manner. These together force an unavoidable gaplessness about εF (dashed line).

  • Table 1 feQBIs in the four Wyckoff-mismatched SGs.

    “Wyckoff multiplicities” denotes the number of lattice points per primitive unit cell required to form an SG symmetric lattice corresponding to one of the Wyckoff positions. νAI and νBand respectively denote the electron fillings consistent with atomic and band insulators, and their discrepancy corresponds to feQBIs.

    Space groupWyckoff multiplicities (5)νAIνBand
    No. 199 (I 213)4, 6, 124ℕ\{4}4ℕ
    No. 214 (I 4132)4, 4, 6, 6, 8, 12, 12, 12, 244ℕ\{4}4ℕ
    No. 220 (I Embedded Image 3d)6, 6, 8, 12, 244ℕ\{4, 8, 20}4ℕ\{4}
    No. 230 (I a Embedded Image d)8, 8, 12, 12, 16, 24, 24, 488ℕ\{8}8ℕ

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/4/e1501782/DC1

    Electron filling in AIs

    Symmetries of SG No. 199

    feQBI tight-binding examples

    Hypothetical structure for spin-orbit coupled hyperkagome material Na3Ir3O8

    Discussions on the SE cut

    Fig. S1. Reproduction of Fig. 1D with different viewing conditions and extra annotation.

    Fig. S2. Energy and entanglement band structure for an alternative feQBI example for SG No. 199.

    Fig. S3. Plot of band gap for the hyperkagome model in eq. S3 at filling ν = 4.

    Fig. S4. Plot of surface band structure against the surface crystal momentum Formula for the model in eq. S3.

    Table S1. List of symmetry elements for SG No. 199.

    Table S2. Spin-quantization axes corresponding to the SG symmetric spin texture.

    Table S3. Symmetry eigenvalues of the irreducible little group representations at high-symmetry momenta.

    Table S4. Transformation of tight-binding sites under the symmetry elements.

    Table S5. A full list of terms in the feQBI tight-binding example given in the text.

    Table S6. Terms in an alternative eight-band feQBI tight-binding example.

    Table S7. Measured structure of Na3Ir3O8 by Takayama et al. (19).

    Table S8. “Symmetry-enriched” hypothetical structure of Na3Ir3O8 in SG No. 214.

  • Supplementary Materials

    This PDF file includes:

    • Electron filling in AIs
    • Symmetries of SG No. 199
    • feQBI tight-binding examples
    • Hypothetical structure for spin-orbit coupled hyperkagome material Na3Ir3O8
    • Discussions on the SE cut
    • Fig. S1. Reproduction of Fig. 1D with different viewing conditions and extra
      annotation.
    • Fig. S2. Energy and entanglement band structure for an alternative feQBI
      example for SG No. 199.
    • Fig. S3. Plot of band gap for the hyperkagome model in eq. S3 at filling v = 4.
    • Fig. S4. Plot of surface band structure against the surface crystal momentum k||
      for the model in eq. S3.
    • Table S1. List of symmetry elements for SG No. 199.
    • Table S2. Spin-quantization axes corresponding to the SG symmetric spin texture.
    • Table S3. Symmetry eigenvalues of the irreducible little group representations at
      high-symmetry momenta.
    • Table S4. Transformation of tight-binding sites under the symmetry elements.
    • Table S5. A full list of terms in the feQBI tight-binding example given in the text.
    • Table S6. Terms in an alternative eight-band feQBI tight-binding example.
    • Table S7. Measured structure of Na3Ir3O8 by Takayama et al. (19).
    • Table S8. “Symmetry-enriched” hypothetical structure of Na3Ir3O8 in SG No.
      214.

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