Band connectivity for topological quantum chemistry: Band structures as a graph theory problem

The conventional theory of solids is well suited to describing band structures locally near isolated points in momentum space, but struggles to capture the full, global picture necessary for understanding topological phenomena. In part of a recent paper [B. Bradlyn et al., Nature (London) 547, 298 (2017)], we have introduced the way to overcome this difficulty by formulating the problem of sewing together many disconnected local $\mathbf{k}·\mathbf{p}$ band structures across the Brillouin zone in terms of graph theory. In this paper, we give the details of our full theoretical construction. We show that crystal symmetries strongly constrain the allowed connectivities of energy bands, and we employ graph theoretic techniques such as graph connectivity to enumerate all the solutions to these constraints. The tools of graph theory allow us to identify disconnected groups of bands in these solutions, and so identify topologically distinct insulating phases.

Figure 1

Two distinct allowed global band structures along the line ${\mathbf{k}}_1\leftrightarrow {\mathbf{k}}_t\leftrightarrow {\mathbf{k}}_2$. ${\rho }_1^1$ and ${\rho }_1^2$ are two distinct representations of $G_{{\mathbf{k}}_1}$, and ${\rho }_2^1$ and ${\rho }_2^2$ are two distinct representations of $G_{{\mathbf{k}}_2}$. All four of these representations subduce the representation ${\sigma }_n$ of the little group $G_{{\mathbf{k}}_t}$ of the line $\left\{{\mathbf{k}}_t\right\}$. Because of this, global energy bands may be connected in two ways. In the band structure (a), the states transforming in representation ${\rho }_1^1$ at ${\mathbf{k}}_1$ connect to the states transforming in the representation ${\rho }_2^1$ at ${\mathbf{k}}_2$. Contrarily, in (b), the states transforming in representation ${\rho }_1^1$ at ${\mathbf{k}}_1$ connect to the states transforming in the representation ${\rho }_2^2$ at ${\mathbf{k}}_2$.

Figure 2

Example of a graph. There are eight nodes labeled $n_1$ through $n_8$, indicated by blue diamonds, black circles, and red squares. There are eight edges, shown as solid black lines connecting pairs of nodes. Nodes in the graph can be placed (amongst other options) in following three partitions: $\{n_1,n_2,n_3,n_8\},\{n_4,n_5,n_6\}$, and $\left\{n_7\right\}$, since there are no edges connecting nodes within each of these sets; the different colors and symbols for the nodes correspond to this partitioning. This graph has two connected components: the first is the subgraph on the left consisting of the nodes $\{n_1,n_2,n_3,n_4,n_5\}$, and the four edges connected to them. The second connected component consists of the three nodes $\{n_6,n_7,n_8\}$ and the four edges connected to them.

Figure 3

Schematic depiction of a trianglelike redundancy. The high-symmetry $\mathbf{k}$ points ${\mathbf{k}}_1, {\mathbf{k}}_2$, and ${\mathbf{k}}_3$ are connected by the symmetry lines ${\mathbf{k}}_4$ and ${\mathbf{k}}_5$. The plane labeled ${\mathbf{k}}_6$ contains all of these $\mathbf{k}$ manifolds. Because of this, the compatibility relations along any path connecting ${\mathbf{k}}_1$ and ${\mathbf{k}}_3$ through ${\mathbf{k}}_6$ provide no additional symmetry constraints beyond what we get from considering the path ${\mathbf{k}}_1\to {\mathbf{k}}_4\to {\mathbf{k}}_2\to {\mathbf{k}}_5\to {\mathbf{k}}_3$. We can thus safely neglect the manifold ${\mathbf{k}}_6$ in our construction of compatibility graphs.

Figure 4

The two connectivity subgraphs shown here are trivially isomorphic if the nodes labeled ${\rho }_i$ correspond to the same little group representations.

Figure 5

Permissible and nonpermissible crossings of edges in connectivity graphs. The blue stars and circles in the ${\mathbf{k}}_2$ partition correspond to two different irreps of $G_{{\mathbf{k}}_2}$. (a) Shows a nonpermissible crossing: when mapped back to a band structure, the energy bands corresponding to the blue circle irreps must cross, and this crossing is not protected. It will generically either gap or move away from the high-symmetry line. On the other hand, (b) shows a permissible crossing corresponding to a protected crossing of energy bands carrying different irreps of $G_{{\mathbf{k}}_2}$.

Figure 6

Connectivity subgraphs along the $\mathrm{\Gamma }\leftrightarrow \mathrm{\Lambda }\leftrightarrow P$ line in (a) SG $I432$ (211) and (b) $I{4}_{1}32$ (214). In each case, there is only one unique subgraph.

Figure 7

Maximal Wyckoff positions for space group $P4mm$ (99), projected along the $\widehat{z}=\left[001\right]$ direction. The blue star indicates the $1a$ position at the Bravais lattice sites, the red diamond indicates the $1b$ position at the center of the unit cell, and the black circles denote the $2c$ Wyckoff position at the middle of the edges. Because space group $P4mm$ (99) has no symmetries constraining $z$, the coordinates of these positions extend in the $z$ direction (perpendicular to the plane of the page).

Figure 8

Visual depiction of the two different disconnected connectivity graphs for space group $P4mm$ (99), corresponding to the Laplacian matrices in Eqs. (46) and (47). In the graph on the left, the ${\overline{\mathrm{\Gamma }}}_6$ little group representation at $\mathrm{\Gamma }$ is connected to the ${\overline{M}}_6$ representation at $M$, while in the graph on the right the ${\overline{\mathrm{\Gamma }}}_7$ little group representation at $\mathrm{\Gamma }$ is connected to the ${\overline{M}}_6$ representation at $M$.

Figure 9

Tight-binding band structure for $s$ orbitals at the $2f$ position in SG $P4/mmm$ (123) in two dimensions. (a) Shows the band structure with no spin-orbit coupling, which yields a semimetal with a gapless point at $M$. (b) Shows the spectrum with nonzero SOC. The band structure is fully gapped.

Figure 10

SG $P4/mmm$ (123) tight-binding model on a finite slab of 50 unit cells. (a) Shows the full slab band structure, with two pairs of counterpropagating edge modes clearly visible in the bulk gap. Because the total centers of charge of orbitals at the $2f$ position are not on Bravais lattice sites, our boundary conditions break inversion symmetry. (b) Shows the surface density of states as a function of energy and surface momentum, showing one pair of counterpropagating edge modes. The other pair of edge modes is on the other surface of the slab (not shown).