# Symmetry-based Indicators of Band Topology in the 230 Space Groups

## TL;DR

This paper develops a comprehensive framework to identify symmetry-based indicators of band topology across all 230 space groups, aiding the discovery of new topological materials.

## Contribution

It introduces a systematic method to reveal all symmetry-based topological indicators in all space groups, extending previous results and uncovering hidden topological phases.

## Key findings

- Unified approach for all 230 space groups
- Identification of previously hidden topological phases
- Framework for discovering new topological materials

## 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.