Topology in shallow-water waves: a violation of bulk-edge correspondence

TL;DR

This paper investigates the topological properties of the rotating shallow-water model, revealing a boundary-condition-dependent edge mode count that violates the traditional bulk-edge correspondence due to scattering amplitude anomalies.

Contribution

It demonstrates a violation of bulk-edge correspondence in a topological fluid model caused by boundary condition effects and scattering theory insights.

Findings

Edge mode count depends on boundary conditions

Violation of bulk-edge correspondence explained

Scattering amplitude anomalies cause mismatch

Abstract

We study the two-dimensional rotating shallow-water model describing Earth's oceanic layers. It is formally analogue to a Schr\"odinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson's theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, ultimately responsible for the violation.