# Delocalization of edge states in topological phases

## TL;DR

This paper investigates how the delocalization of edge states occurs in topological phases when the indirect energy gap closes, showing that topological invariants do not guarantee localized edge modes.

## Contribution

It demonstrates that closing the indirect gap in lattice models leads to delocalized edge states despite the persistence of topological invariants.

## Key findings

- Edge states can become delocalized when the indirect gap closes.
- Topological invariants do not necessarily imply localized edge modes.
- The inverse participation ratio effectively measures localization.

## Abstract

The presence of a topologically non-trivial discrete invariants implies the existence of gapless modes in finite samples, but it does not necessarily imply their localization. The disappearance of the indirect energy gap in the bulk generically leads to the absence of localized edge states. We illustrate this behavior in two fundamental lattice models on the single-particle level. By tuning a hopping parameter the indirect gap is closed while maintaining the topological properties. The inverse participation ratio is used to measure the degree of localization.