# The hybrid-order topology of weak topological insulators

## TL;DR

This paper reveals a new class of weak topological insulators with higher-order topological features protected by rotation symmetry, featuring surface Dirac cones and hinge modes, linked to a hidden crystalline invariant.

## Contribution

It introduces hybrid-order weak topological insulators with rotation symmetry, uncovering a hidden crystalline invariant responsible for their topological phase.

## Key findings

- Surface Dirac cones on all surfaces.
- Hinge modes persist after surface gap opening.
- Topological phase linked to a hidden invariant.

## Abstract

We consider weak topological insulators with a twofold rotation symmetry around the dark direction, and show that these systems can be endowed with the topological crystalline structure of a higher-order topological insulator protected by rotation symmetry. These hybrid-order weak topological insulators display surface Dirac cones on all surfaces. Translational symmetry breaking perturbations gap the Dirac cones on the side surfaces leaving anomalous helical hinge modes behind. We also prove that the existence of this topological phase comes about due to a hidden crystalline topological invariant of quantum spin-Hall insulators that can neither be revealed by symmetry indicators nor using Wilson loop invariants. Considering the minimal symmetry requirements, we anticipate that our findings could apply to a large number of weak topological insulators.