# Topological protection by local support symmetry and destructive interference

## TL;DR

The paper shows that topological features can be protected by symmetries that exist only in part of a system, not requiring full system-wide symmetry.

## Contribution

The novel contribution is the concept of local support symmetries that can protect topological phases even when global symmetry is broken.

## Key findings

- Local support symmetries can protect topological features despite symmetry-breaking couplings.
- Destructive interference of Bloch wave functions is crucial for this protection.
- A fluorinated biphenylene network demonstrates a material realization of this concept.

## Abstract

Conventionally, symmetry-protected topological phases and band crossings are protected by global symmetries acting on the entire system. Here, we show that symmetries preserved only on a partial region of a system, termed local support symmetries, can protect topological features of the full system, even in the presence of symmetry-breaking couplings. We establish a unified framework by deriving explicit conditions for such protection in both insulating and metallic phases and show that destructive interference of Bloch wave functions plays a key role. Using representative tight-binding models, we demonstrate band crossings and topological bands protected by local support crystalline and time-reversal symmetries, and further present a realistic material realization in a fluorinated biphenylene network, where a band crossing is protected by a local support C2 symmetry.

Symmetry-protected topological phases are conventionally considered to be protected by global symmetries. Here, the authors demonstrate that topological protections can persist even when symmetries are preserved only within a partial region of the system, which they term local support symmetries.