Topological Classification of Crystalline Insulators through Band Structure Combinatorics

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented match the mathematical structure underlying the topological classification of these crystals in terms of $K$-theory and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify all allowed topological phases of spinless particles in crystals in class $A$. Employing this classification, we study transitions between topological phases within class $A$ that are driven by band inversions at high-symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure and to identify whether or not they give rise to intermediate Weyl semimetallic phases.

Popular Summary

Topological insulators are exotic materials that are electrical insulators in their interior but can conduct electricity on their surface, and their discovery has fundamentally changed our understanding of how phases of matter may be organized. Most phases of matter are categorized by the symmetries that they break. Crystals break translational symmetry, magnets break rotational symmetry, and so on. Topological insulators, however, show that some phases can be distinct even though their symmetries are equal. Researchers have come up with a classification scheme—called the tenfold way—which allows for the categorization of topological phases depending on some general properties, such as whether or not they have time-reversal or particle-hole symmetry. We complement this categorization by providing a method for listing all possible topologically distinct phases of matter that do not have external symmetries but do have the types of internal (or lattice) symmetries that appear in the atomic arrangements of real solid materials.

We explicitly list all possible phases in two-dimensional materials and provide an intuitive and easily applicable method for identifying the phases possible within a given lattice type in any dimension. Our method matches the known results based on the mathematically involved predictions of $K$-theory, which is known to be a rigorous way of identifying all possible topological phases. It thus provides insight into this mathematical arena based on a physical understanding of topological band structures. We also show how our method can be used to study the transitions between topological phases and predict whether they will result in topologically protected Weyl semimetals.

This new classification can now be used to guide the search for new types of topological materials and related edge modes or exotic intermediate phases.

Figure 1

The fundamental domain (shaded red) $\mathrm{\Omega }$ of the first Brillouin zone. It contains only points that are not related to each other by transformations in the point group $D_4$, as described by Eq. (2). High-symmetry lines are indicated in red, and high-symmetry points are in black.

Figure 2

Possible effects of a band inversion at $\mathrm{\Gamma }$ on band structure topologies with $p4mm$ symmetry. Red lines are electronic bands, which are flattened for clarity. The dashed blue line indicates the Fermi energy. The eigenvalues along a high-symmetry line under the associated reflection symmetry are indicated with $\pm$. (a) A topologically insulating state with one valence band and one conduction band. The representations, or $\pm 1$ eigenvalues, along high-symmetry lines are indicated. Representations at high-symmetry points are indicated for the valence band only, as the conduction band is in the trivial representation at all high-symmetry points. This configuration is gapped everywhere and has $N_{\pm }^{l_1}=0$. (b) Upon an inversion at $\mathrm{\Gamma }$, two bands will cross along $l_1$, creating a Weyl cone. The crossing is characterized by $N_{\pm }^{l_1}=\pm 1$. Note that along $l_3$, a crossing has been avoided. (c) A topologically insulating state with a valence band that is odd on both $l_1$ and $l_3$. (d) An inversion at $\mathrm{\Gamma }$, in this case, resulting in Weyl nodes along both $l_1$ and $l_3$, characterized by $N_{\pm }^{l_1}=\pm 1=-N_{\pm }^{l_3}$.

Figure 3

Inverting a nondegenerate band with a doubly degenerate one at fixed filling. (a) A topologically insulating state with two valence bands and one conduction band. The valence bands are degenerate at $\mathrm{\Gamma }$, and $N_{\pm }^{l_i}=0$ for all $l_i$. (b) A band inversion at $\mathrm{\Gamma }$ at fixed filling requires the number of filled bands after the inversion to still be two everywhere. This implies that, at $\mathrm{\Gamma }$, a degeneracy must appear at the Fermi level. This intermediate phase is characterized by $N_{+}^{l_1}=-N_{+}^{l_3}=1$, $N_{-}^{l_1}=N_{-}^{l_3}=0$.

Figure 4

Inverting two doubly degenerate bands. (a) A topologically insulating state with two valence bands and two conduction bands, which are both degenerate at $\mathrm{\Gamma }$. (b) After a band inversion at $\mathrm{\Gamma }$, there will be four band crossings on each high-symmetry line. Two of them will be avoided and not drawn, whereas the other two constitute Weyl nodes. (c) Zooming in on the boxed region in diagram (b), the two opposite Weyl nodes along a single high-symmetry line can be moved towards each other and annihilated. A direct transition between the original and final topologically insulating states is therefore possible. Notice that the final state is the same as the original one. This agrees with the fact that the integers $N_{+}$ and $N_{-}$ are all zero in both the initial and final configurations.

Figure 5

Sketch of the relations between high-symmetry points imposed by high-symmetry lines. A set of valence bands is shown between two generic high-symmetry points $X$ and $Y$, which are left invariant under the symmetry transformations contained in the little co-groups $G^X$ and $G^Y$. The irreducible representations of $G^X$ and $G^Y$ may be labeled as $X_i$ and $X_j$. Each of the bands needs to fall within one of these representations at the corresponding high-symmetry points. The representations on $X$ and $Y$, however, need to be compatible with the representations ${\mathrm{\Lambda }}_j$ of the high-symmetry line connecting $X$ and $Y$. The remaining set of independent integers indicating how many bands are in which representations at the high-symmetry points finally determines the number of possible topologically distinct configurations of the set of valence bands, matching the abstract $K$-theory classification.

Figure 6

Fundamental domain of the octahedral group $O_h$ (shaded red). The red high-symmetry lines connect the high-symmetry points $\mathrm{\Gamma }$, $M$, $X$, and $R$.

Figure 7

Representations of $G^{\alpha }$ along $\alpha$ induce representations of $G^{{\mathbf{k}}_i}$ at ${\mathbf{k}}_i$. The shaded paths indicate other possible fixed-point sets in $\mathcal{M}$.

Figure 8

Fundamental domain $\mathrm{\Omega }$ of the octahedral group ${\mathbf{Z}}_2\times S_4$ (shaded red). The red lines are the high-symmetry lines $\mathrm{\Lambda }$, $\mathrm{\Delta }$, $\mathrm{\Sigma }$, $S$, $T$, and $Z$.