# Classification of crystalline insulators without symmetry indicators: atomic and fragile topological phases in twofold rotation symmetric systems

## TL;DR

This paper introduces a new classification scheme for two-dimensional crystalline insulators with twofold rotation symmetry, identifying fragile topological phases beyond traditional symmetry indicator methods, with implications for novel 2D materials.

## Contribution

It presents the first classification of topological crystalline phases that are undetectable by symmetry indicators, focusing on fragile phases in twofold rotation symmetric systems.

## Key findings

- Identified three $\ ext{Z}_2$ topological invariants via nested Berry phases.
- Constructed microscopic models exhibiting fragile topological phases.
- Highlighted the importance of fragile phases in materials like twisted bilayer graphene.

## Abstract

Topological crystalline phases in electronic structures can be generally classified using the spatial symmetry characters of the valence bands and mapping them onto appropriate symmetry indicators. These mappings have been recently applied to identify thousands of topological electronic materials. There can exist, however, topological crystalline non-trivial phases that go beyond this paradigm: they cannot be identified using spatial symmetry labels and consequently lack any classification. In this work, we achieve the first of such classifications showcasing the paradigmatic example of two-dimensional crystals with twofold rotation symmetry. We classify the gapped phases in time-reversal invariant systems with strong spin-orbit coupling identifying a set of three $\mathbb{Z}_2$ topological invariants, which correspond to nested quantized partial Berry phases. By further isolating the set of atomic insulators representable in terms of exponentially localized symmetric Wannier functions, we infer the existence of topological crystalline phases of the fragile type that would be diagnosed as topologically trivial using symmetry indicators, and construct a number of microscopic models exhibiting this phase. Our work is expected to have important consequences given the central role fragile topological phases are expected to play in novel two-dimensional materials such as twisted bilayer graphene.