Research ArticleCONDENSED MATTER PHYSICS

Topological states from topological crystals

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Science Advances  18 Dec 2019:
Vol. 5, no. 12, eaax2007
DOI: 10.1126/sciadv.aax2007
  • Fig. 1 Cell complex structure for space group P1̄ (#2).

    (A) The AU 0 < x, y < 1 and 0<z<12. (B) The symmetry-inequivalent 2-cells (colored faces), 1-cells (bold lines), and 0-cells (red dots). The other cells can be obtained from these by acting with symmetry operations.

  • Fig. 2 Topological crystals in space group P42212 (#94).

    (A) The symmetry-inequivalent 2-cells (ei=1,2,3,42) and 1-cells (ei=1,2,3,4,5,61) are represented by colored faces and bold lines, respectively. Here, the lattice constants are set to 1, the unit cell is given by 0 ≤ x, y, z < 1, and the AU is given by 0<x,y,z<12. (B to D) The three independent ℤ2 topological crystal generators, where only 2-cells decorated with 2dTIs are shown. (C) and (D) are layer constructions, whereas (B) is not.

  • Fig. 3 Illustrations of the topological crystals beyond layer construction.

    (A) to (L) show these states for space groups Pnn2 (#34), Pnnn (#48), P42 (#77), P42/n (#86), P4222 (#93), P42212 (#94), P42cm (#102), P4̄n2 (#118), P42/nnm (#134), Pn3̄ (#201), P4232 (#208), and Pn3̄m (#224), respectively. Inequivalent 2-cells are represented by different colors. The topological invariants of these topological crystals and the coordinates of the plotted 2-cells can be found in the Supplementary Materials.

  • Fig. 4 Illustration of the effects of the equivalence operation in point group 1̄ without time-reversal symmetry.

    (A) The 2-cells: e12:z=0,x>0 and e22:z=0,x<0. (B) The equivalence operation. When a bubble made of 2D Chern insulator with Chern number 1 is created in z > 0, another Chern bubble with Chern number −1 in z < 0 will be created because of the inversion symmetry. The arrows represent the orientation of the Chern number. We enlarge these two bubbles until they join with the z = 0 plane; then, the Chern number of the state on z = 0 will be subducted by 2.

  • Table 1 Classifications C of TCIs for noninteracting electrons with time-reversal symmetry, spin-orbit coupling, and crystalline point group symmetry.

    N/A denotes a trivial classification. Full classifications including strong topological insulators can be easily obtained from this table by using the group extension rules given in Results and are also provided as a table in the Supplementary Materials.

    Point groupTCI classification
    1N/A
    1¯2
    22
    m
    2/mℤ × ℤ2
    22222
    mm22
    mmm3
    42
    4¯2
    4/mℤ × ℤ2
    42222
    4mm2
    4¯2mℤ × ℤ2
    4/mmm3
    3N/A
    3¯2
    322
    3m
    3¯mℤ × ℤ2
    62
    6¯
    6/mℤ × ℤ2
    62222
    6mm2
    6¯2m2
    6/mmm3
    23N/A
    m3¯
    4322
    4¯3m
    m3¯m2
  • Table 2 TCI classifications C for noninteracting electrons with time-reversal symmetry and spin-orbit coupling, given for all space groups.

    N/A denotes a trivial classification. Full classifications including strong topological insulators can be easily obtained from this table by using the group extension rules given in Results and are also provided as a table in the Supplementary Materials.

    12341238123121 × 22161220122
    224422 × 22822212222162 × 22202
    3244322832 × 221235163222032
    423442 × ℤ284 × 22124 × 23164 × 22204ℤ × ℤ2
    52345238523125 × 23165222052
    62 × 2246 × 22862312623166 × 2220622
    72347687 × 221273 × ℤ21672220722
    8 × 2248248822128 × 221682220822
    92249 × 2489241292 × 2216922092
    102 × 23502490231302317022102
    11 × 23513×2291231313 × ℤ21712221122
    12 × 23522392221322 × 22172222122
    132453 × 2393241332317322132
    142354249423134 × 23174221422
    1523552 × 229523135 × 221752 × ℤ2215ℤ × ℤ2
    1625562396221362 × ℤ2176ℤ × ℤ2216
    172457 × 239723137 × 2217723217ℤ × ℤ2
    182358 × 229823138 × 22178222182
    1922592 × 22993 × ℤ21393 × ℤ2179222192
    20236023100 × 221402 × 22180232202
    21246123101 × 22141 × 22181232213
    222462 × 22102 × 22142231822222222
    2323632 × 221032314321832 × ℤ2223ℤ × ℤ2
    242364 × 2310422144218422224 × 22
    254 × ℤ2654 × ℤ21052 × ℤ21452185ℤ × ℤ22252
    262 × 2266 × 23106221462186ℤ × ℤ2226ℤ × ℤ2
    2724672 × 231072 × ℤ2147221873227ℤ × ℤ2
    28 × 236824108 × 2214822188ℤ × ℤ222822
    2923693×22109ℤ × ℤ21492218932292 × ℤ2
    302370231102215022190ℤ × ℤ223022
    31 × 22713 × ℤ2111 × 23151221914
    322372 × 231122315222192 × 22
    33227324113 × 22153221932 × ℤ2
    3423742 × 2211422154221942 × ℤ2
    352 × 2275231152 × 22155221952
    36 × 22762211623156ℤ × ℤ2196N/A
    3723772311723157ℤ × ℤ21972
    383 × ℤ27822118231582198N/A
    39 × 237922119 × 2215921992
    40 × 22802212023160ℤ × ℤ22002

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/12/eaax2007/DC1

    Fig. S1. Illustration of the 2-cells e12 and e22 and the 1-cell e1 used to discuss the effect of gluing conditions on MTCIs.

    Fig. S2. The 3-cells e13 and e13 (e23 and e23) meet at the 2-cell e12 (e22).

    Fig. S3. Cross section through e1 and the 2-cells that meet at e1, in case (2b.ii), where e1 is contained in a C3v axis.

    Fig. S4. Illustration of the effects of the gluing condition for ℤ2 TCIs for two representative space groups.

    Table S1. Topological crystals beyond layer construction.

    Table S2. Classification 𝒞full of STIs and TCIs with time-reversal symmetry and spin-orbital coupling for all point groups.

    Table S3. Classification 𝒞full of STIs and TCIs with time-reversal symmetry and spin-orbital coupling for all space groups.

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Illustration of the 2-cells e12 and e22 and the 1-cell e1 used to discuss the effect of gluing conditions on MTCIs.
    • Fig. S2. The 3-cells e13 and e13 ( e23 and e23) meet at the 2-cell e12 ( e22).
    • Fig. S3. Cross section through e1 and the 2-cells that meet at e1, in case (2b.ii), where e1 is contained in a C3v axis.
    • Fig. S4. Illustration of the effects of the gluing condition for ℤ2 TCIs for two representative space groups.
    • Table S1. Topological crystals beyond layer construction.
    • Table S2. Classification Cfull of STIs and TCIs with time-reversal symmetry and spin-orbital coupling for all point groups.
    • Table S3. Classification Cfull of STIs and TCIs with time-reversal symmetry and spin-orbital coupling for all space groups.

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