Quantitative mappings between symmetry and topology in solids

TL;DR

This paper develops comprehensive mappings from symmetry data to topological invariants in solids, simplifying the identification of topological states in materials with various symmetries.

Contribution

It provides explicit, exhaustive mappings from symmetry data to topological invariants for all gapped band structures with time-reversal symmetry and any of the 230 space groups.

Findings

Mapped symmetry data to topological invariants for all space groups

Simplified the process of identifying topological states in materials

Enabled straightforward search for topological invariants using provided tables

Abstract

The study of spatial symmetries was accomplished during the last century, and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here, we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided.