Graph theory data for topological quantum chemistry

Topological phases of noninteracting particles are distinguished by the global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties that are local (at high-symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [Bradlyn et al., Nature (London) 547, 298 (2017)] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain “elementary” band structures highlighted in the aforementioned reference, and thus we identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed bandrep program on the Bilbao Crystallographic Server to access the results of our computation.

Figure 1

Partial view ($0\le k_x,k_y,k_z\le 1/2$ region) of the first Brillouin zone of the space group $P4/ncc$ (130). The special $\mathbf{k}$-vectors of Table 1: the points of maximal symmetry $\mathrm{\Gamma },Z,M,A,R,X$; lines $\mathrm{\Lambda },V,W,\mathrm{\Sigma },S,\mathrm{\Delta },U,Y,T$; and planes $D,E,C,B,F$ have been indicated.

Figure 2

Constraints on energy bands due to glide-reflection symmetry along the $\mathrm{\Gamma }-\mathrm{\Lambda }-Z$ line in space group $P4/ncc$ (130). The points $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ are equivalent, related by a translation by the reciprocal-lattice vector $\mathbf{K}={\mathbf{e}}_3^{*}$. The solid black band carries the representation ${\mathrm{\Lambda }}_4$, while the dashed blue band carries the representation ${\mathrm{\Lambda }}_1$. Inversion symmetry pins the monodromy-enforced band crossing at the $Z$ point.

Figure 3

Example of a graph connected in two different ways. Solid circles are nodes connected by four edges (solid lines). We have two partitions, {$n_1,n_2,n_3$} and {$n_4,n_5$}. We have two ways of connecting the different irreps.

Figure 4

(a) Band structure with two copies of the representation $A_5$ at the point $A$, connected through a line $V$ to distinct representations $M_1$ and $M_2$ at the point $M$. Permuting the representations at $A$ in each case would lead to the same connected elements, since both the left and right subgraphs have the same adjacency matrix. (b) Visual example of fake Weyls. The first two submatrices in Table 10 lead to the connectivity (I), while the last two lead to the connectivity (II). [The crossing of the two blue lines in (II), which correspond to the same representation, gaps out, and so we see that these yield the same connectivity.]

Figure 5

Main input screen for the bandrep program.

Figure 6

Output of bandrep for the elementary band representations in SG $I{2}_{1}3$ (199) without TR. There is one decomposable elementary band representation. It is induced from the two-dimensional ${}^2\overline{E}$ representation of the site-symmetry group of the $12b$ Wyckoff position.

Figure 7

Possible decompositions of the elementary band representation in SG $I{2}_{1}3$ (199) induced from the ${}^2\overline{E}$ representation of the site-symmetry group of the $12b$ maximal Wyckoff position.

Figure 8

Output of bandrep for the physically elementary band representations in SG $I{2}_{1}3$ (199). There are two decomposable physically elementary band representations, induced from the $8a$ and $12b$ maximal Wyckoff position

Figure 9

Decomposition of the elementary band representation in SG $I{2}_{1}3$ (199) induced from the ${}^1\overline{E}{}^2\overline{E}$ physically irreducible representation of the site-symmetry group of the $12b$ maximal Wyckoff position.

Figure 10

Minimal path and associated compatibility tables for SG $I{2}_{1}3$ (199) without TR symmetry.

Figure 11

Minimal path and associated compatibility tables for SG $I{2}_{1}3$ (199) in the presence of TR symmetry.

Figure 12

Output of bandrep for the $2b$ position of SG $P6mm$ (183) with TR symmetry.

Figure 13

Possible decompositions of the physically elementary band representation induced from the ${\overline{E}}_1\uparrow G$ representation of the site-symmetry group of the $2b$ position in SG $P6mm$ (183).