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. 2017 Jun 30;8(1):50.
doi: 10.1038/s41467-017-00133-2.

Symmetry-based indicators of band topology in the 230 space groups

Hoi Chun Po  1   2 , Ashvin Vishwanath  3   4 , Haruki Watanabe  5
Affiliations

Symmetry-based indicators of band topology in the 230 space groups

Hoi Chun Po et al. Nat Commun. .

Erratum in

Abstract

The interplay between symmetry and topology leads to a rich variety of electronic topological phases, protecting states such as the topological insulators and Dirac semimetals. Previous results, like the Fu-Kane parity criterion for inversion-symmetric topological insulators, demonstrate that symmetry labels can sometimes unambiguously indicate underlying band topology. Here we develop a systematic approach to expose all such symmetry-based indicators of band topology in all the 230 space groups. This is achieved by first developing an efficient way to represent band structures in terms of elementary basis states, and then isolating the topological ones by removing the subset of atomic insulators, defined by the existence of localized symmetric Wannier functions. Aside from encompassing all earlier results on such indicators, including in particular the notion of filling-enforced quantum band insulators, our theory identifies symmetry settings with previously hidden forms of band topology, and can be applied to the search for topological materials.Understanding the role of topology in determining electronic structure can lead to the discovery, or appreciation, of materials with exotic properties such as protected surface states. Here, the authors present a framework for identifying topologically distinct band-structures for all 3D space groups.

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Conflict of interest statement

The authors declare no competing financial interests.

Figures

Fig. 1
Fig. 1
Symmetry-based indicators of band topology. a Symmetry labeling of bands in a 1D inversion-symmetric example. k 0 = 0, π are high-symmetry momenta, where the bands are either even (+) or odd (−) under inversion symmetry (orange diamonds). From a symmetry perspective, a target set of bands (purple and boxed) separated from all others by band gaps can be labelled by the multiplicities of the two possible symmetry representations, which we denote by the integers nk0±. Note that such labeling is insensitive to the detailed energetics within the set. In addition, the set is also characterized by the number of bands involved, which we denote by ν. Altogether, the set is characterized by five integers, which are further subjected to the constraints ν=n0++n0-=nπ++nπ-. b Symmetry labels like those described in a can be similarly defined for systems symmetric under any of the 230 space groups in three dimensions. Using such labels, one can reinterpret the set of band structures as an Abelian group. This is schematically demonstrated through the two labels ν and n α, which organize the set of all possible band structures into a two-dimensional lattice. Note that the dimensionality of this lattice is given by the number of independent symmetry labels, and is a property of the symmetry setting at hand. Organized this way, the band structures corresponding to atomic insulators, which are trivial by our definition, will generally occupy a sublattice. Any band structure that does not fall within this sublattice necessarily possesses nontrivial band topology
Fig. 2
Fig. 2
Examples of topological band structures. a–c A representation-enforced quantum band insulator of spinful electrons with time-reversal and inversion symmetries, dubbed the “doubled strong TI”. a Using the Fu-Kane parity criterion, the strong and weak Z2 indices can be computed from the the parities of the occupied bands, which we indicate by ± at the eight time-reversal invariant momenta. Shown are the parities of one state from each Kramers pair for a doubled strong TI with four filled bands. b The entanglement spectrum at a spatial cut, parallel to the x–y plane and containing an inversion center, features two Dirac cones at Γ, , . Such Dirac cones are known to possess integer-valued charges under the inversion symmetry, and we denote the positively charged and negatively charged cones, respectively, by blue and red. c Inversion-symmetric atomic insulators feature entanglement surface Dirac cones in general, but their presence depends on the arbitrary choice of the cut. We find that the possible Dirac-cone arrangement arising from atomic insulators can only be a linear combination of four basic configurations, illustrated as a sum with the integral weights m i. The arrangement in b cannot be reconciled with those in c, confirming the nontriviality of the doubled strong TI. d, e Example of a lattice-enforced semimetal for spinful electrons with time-reversal symmetry. d We consider a site (red sphere) under a local environment (beige) symmetric under the point group T, and suppose the relevant local energy levels form the four-dimensional irreducible representation, which is half-filled (boxed). e When the red site sits at the highest-symmetry position of space group 219, the specified local energy levels and filling gives rise to a half-filled eight-band model (each band shown is doubly degenerate). Such (semi-)metallic behavior is dictated by the specification of the microscopic degrees of freedom in this model
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