# Anomalous bulk-edge correspondence in continuous media

## TL;DR

This paper reveals that in continuous media, the bulk-edge correspondence can appear violated due to hidden ghost edge modes, and provides a scattering theory-based method to correctly identify all edge modes, restoring the expected topological relation.

## Contribution

It introduces a formalism using scattering theory to detect hidden ghost edge modes in continuous media, resolving the apparent violation of bulk-edge correspondence.

## Key findings

- Ghost edge modes are responsible for the apparent violation.
- The formalism successfully detects all edge modes in models.
- Restoring the bulk-edge correspondence clarifies topological properties.

## Abstract

Topology plays an increasing role in physics beyond the realm of topological insulators in condensed mater. From geophysical fluids to active matter, acoustics or photonics, a growing family of systems presents topologically protected chiral edge modes. The number of such modes should coincide with the bulk topological invariant (e.g. Chern number) defined for a sample without boundary, in agreement with the bulk-edge correspondence. However this is not always the case when dealing with continuous media where there is no small scale cut-off. The number of edge modes actually depends on the boundary condition, even when the bulk is properly regularized, showing an apparent paradox where the bulk-edge correspondence is violated. In this paper we solve this paradox by showing that the anomaly is due to {ghost} edge modes hidden in the asymptotic part of the spectrum. We provide a general formalism based on scattering theory to detect all edge modes properly, so that the bulk-edge correspondence is restored. We illustrate this approach through the odd-viscous shallow-water model and the massive Dirac Hamiltonian, and discuss the physical consequences.