Exhaustive list of topological hourglass band crossings in 230 space groups

Lin Wu, Feng Tang, and Xiangang Wan

Phys. Rev. B 102, 035106 – Published 2 July 2020

Abstract

Topological semimetals with band crossings (BCs) near the Fermi level have attracted intense research activity in the past several years. Among various BCs, those enforced by an hourglasslike connectivity pattern, which are located just at the vertex in the neck of an hourglass and thus called hourglass BCs (HBCs), show interesting topological properties and are intimately related with the space group symmetry. Through checking compatibility relations in the Brillouin zone, we list all possible HBCs for all 230 space groups by identifying positions of HBCs as well as the compatibility relations related with the HBCs. The HBCs can be compatible with conventional topological BCs such as Dirac and Weyl fermions and, based on our exhaustive list, the dimensionality and degeneracy of the HBCs can be quickly identified. It is also found that the HBCs can be classified into two categories: one contains essential HBCs which are guaranteed to exist, while the HBCs in the other category may be tuned to disappear. Our results can help in efficiently predicting hourglass semimetals combined with first-principles calculations as well as studying transitions among various topological crystalline phases.

Received 27 January 2020
Revised 19 May 2020
Accepted 8 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.035106

Physics Subject Headings (PhySH)

Electronic structure Topological materials

Symmetries in condensed matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Lin Wu, Feng Tang ^*, and Xiangang Wan

National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

^*Corresponding author: fengtang@nju.edu.cn

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Vol. 102, Iss. 3 — 15 July 2020

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Figure 1
Classification of the hourglass band structures: (a)–(e) correspond to hourglass BCs of types a–e, respectively. R, B denote two (sets of) $k$ vectors in the BZ which must be HSP or HSL while X, which can be HSL or HSPL, connects R and B. Energy bands from R to X split according to the compatibility relations. For stable band crossings as shown in the neck of hourglass, X must allow two different irreps, as indicated by black solid and purple dotted lines. For types a–c, the numbers of additional involved irreps other than the two irreps in then neck for the bottom and top bands are 0, 1, 2, respectively, as indicated by different colors and styles of lines. It is easy to find that these hourglass band structures are stable against switching band orderings of R or B, owing to different irreps in the bottom and top bands. However, the remaining two hourglass structures in d and e can be tuned to gapped as shown in the last panel since the bottom and top bands share the same irrep. Note that for types a–c, once for R and B, all possible splitting patterns in X are and can only be those as displayed, the hourglass band connectivity must exist since the partner switch of any Bloch states have to happen to satisfy the compatibility relations, thus they are essential.
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Figure 2
When two different hourglass structures both of type a (see Fig. 1), called type a+type a hourglass BC, have existed, they can still be turned to be gapped through tuning the band orderings. Here we show a concrete example in SG 113 when considering TRS and neglecting SOC, any point in the HSL V can own three two-dimensional irreps (labeled V1,2; V3,3; and V4,4; indicated by dashed green, solid green, and solid red lines, respectively). The HSP A can also host three 2-dimensional irreps (labelled by A1,3, A2,4 and A5). The compatibility relations between A and V are A1,3 $\to V$ 3,3, A2,4 $\to V$ 4,4 and A5 $\to V$ 1,2. The HSPL C could only own two one-dimensional irreps: C1 and C2. Consider four bands (all are twofold degenerate) crossing as in (a). Thus, when considering the path $C_{1}$ in C connecting A and $V_{1}$ in V, there would be two hourglass structures both of type a shown in the left panel of (b) due to the compatibility relations for the energy levels in order at A and $V_{1}$ : A5 $\to C$ 1+C2, A5 $\to C$ 1+C2, A1,3 $\to$ 2C1, A2,4 $\to 2 C 2$ and V4,4 $\to$ 2C2, V3,3 $\to$ 2C1, V1, $2 \to C$ 1+C2, V1, $2 \to C$ 1+C2. However, for another path $C_{2}$ connecting A and $V_{2}$ , the hourglasses disappear since the energy levels at $V_{2}$ allow the direct connection of the four energy levels, according to the compatibility relations.
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Figure 3
(a) The BZ of base-centred orthorhombic lattice, where the labels of inequivalent HSPs and HSLs are given, following the notation adopted from the Bilbao server [50]. (b) The BZ of primitive orthorhombic lattice, where the labels of inequivalent HSPs and HSLs are given, following the notation adopted in the Bilbao server [50]. (c)The BZ of the trigonal lattice where we label all the inequivalent HSPs and HSLs following the Bilbao Server [50]. (d) Electronic band structure of ${AsPb}_{2} {Pd}_{3}$ (SG 36) by first principles calculations where S-D-R line could always host hourglass nodal points. S and R are two HSPs and D is HSL. Two such nodal points are circled by dashed curves almost near the Fermi level. (e)The first-principles calculated electronic band structure of ${Sr}_{2} {Bi}_{3}$ (SG 52) with the hourglass nodal point indicated by orange dashed circle. (f) The principle-calculated electronic band structure of ${ReO}_{3}$ (SG 182) with three Weyl hourglass points indicated by dashed circles. Note that, for the highest circle, there exist two BCs, one (denoted by black arrow) is caused by hourglass structure and robust while the other one (denoted by the red arrow) is not robust and can be tuned to disappear as shown in the inset.
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Figure 4
(a) The electronic band structure of ${Ba}_{2} {ReO}_{5}$ where $G^{'}$ is the middle point of UX. Note that the bands in X- $G^{'}$ are always fourfold degenerate since only one four-dimensional irrep is allowed in HSL G. In fact, any path (even not a straight line) connecting S and any point in HSL G could host essential hourglass BC. We highlight six hourglass BCs in the band structure with two of them enlarged, shown in the right-side insets. (b) The expected hourglass nodal loop (the white loop) around S in the L plane is verified by our first-principles calculations.
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