Diagnosis for Nonmagnetic Topological Semimetals in the Absence of Spin-Orbital Coupling

Topological semimetals are under intensive theoretical and experimental studies. The first step of these studies is always the theoretical (numerical) predication of one or several candidate materials, based on first-principles numerics. In these calculations, it is crucial that all topological band crossings, including their types and positions in the Brillouin zone, are found. While band crossings along high-symmetry lines, which are routinely scanned in numerics, are simple to locate, the ones at generic momenta are notoriously time-consuming to find and may be easily missed. In this paper, we establish a theoretical scheme of diagnosis for topological semimetals where all band crossings are at generic momenta in systems with time-reversal symmetry and negligible spin-orbital coupling. The scheme uses only the symmetry (inversion and rotation) eigenvalues of the valence bands at high-symmetry points in the Brillouin zone as input and provides the types (lines or points), topological charges, numbers, and configurations of all robust topological band crossings, if any, at generic momenta. The nature of the new diagnosis scheme allows for full automation and parallelization and paves the way to high-throughput numerical predictions of topological semimetals.

Popular Summary

Topological materials are now under intensive research efforts for their unique properties such as their exotic surface behavior and the anomalous response of the interior-to-external perturbations. Identifying topological materials among the vast database of real materials is, however, a formidable task because the requisite quantum numbers (known as topological invariants) are notoriously difficult to compute. Here, we develop a new diagnosis scheme for all topological semimetals in which there is weak spin-orbit coupling.

We establish, for each existing lattice structure, a complete set of equations relating the “symmetry data” and the “topology data.” Symmetry data are how the wave functions of the valence bands transform under the symmetries of the lattice. Topology data are the set of all topological invariants that, in the absence of spin-orbital coupling, describe the number and positions of all topological nodes in the momentum space. The new scheme circumvents the difficult problem of numerically computing the topological invariants, and it reduces the diagnosis of a candidate for topological material to computing the symmetry data, which can be readily extracted from modern first-principles calculations.

To date, the hunt for topological materials has been mostly ad hoc, and its success has critically depended on the researchers’ experience and knowledge in materials science. Our theory, based on rigorous mathematics, serves as a basis for building a fully automated diagnosis program for all topological materials, which requires no human intervention or knowledge in materials science or topological physics.

Figure 1

The loops on which Berry phases are considered in this paper. The plane in (a) can be any 2D submanifold of the 3D BZ in a system having inversion and the plane in (a) and (b) can be any $k$ slice in a system having a twofold or fourfold axis. (a) A loop enclosing half of the BZ, passing four TRIMs on its way, denoted by ${\mathbf{K}}_{1,2,3,4}$. (b) A loop enclosing a quarter of the BZ, passing all four TRIMs. (c) A loop particularly considered in a simple tetragonal lattice for space group 130.

Figure 2

(a) Two nodal rings that are contributed by the two copies of the (0001) state have two loops at the same position (deliberately separated for distinction), each crossing the $k_3=0$ plane. (b) Adding a hybridization between the two copies can open a full gap on the $k_3=0$ plane, which results in a reconnection of the nodal structure, such that, on each side of $k_3=0$, there is a nodal ring with nontrivial ${\mathbb{Z}}_2$-monopole charge.

Figure 3

A loop considered in a body-centered tetragonal lattice for space group 87.

Figure 4

Loops along which the Berry phases are used to define indicators in noncentrosymmetric space groups (a) 27, (b) 37, and (c) 103 (see Table 3).

Figure 5

Minimal configurations of the Weyl points for each nonzero set of indicators in various space groups. Red lines are rotation axes, gray lines the intersection between vertical glide planes and the $k_z=\pi$ plane, red dots $S_4$ centers, and “$w$” stands for Weyl, where different colors of $w$ mean opposite monopole charge. (a)–(h) are for space groups with only one ${\mathbb{Z}}_2$ indicator: (a) 3, (b) 27, (c) 37, (d) 75 and 77, (e) 82, (f) 103, (g) 168, 171, and 172, and (h) 184. (i)–(k) are the minimal configurations for group 81 having indicator sets $({\omega }_2^0,{\omega }_2^{\pi })=(1,0),(0,1),(1,1)$, respectively.