Topology in shallow-water waves: A spectral flow perspective

TL;DR

This paper investigates the topological properties of shallow-water wave models, introducing a novel spectral flow-based edge index that resolves anomalies and links boundary phenomena to bulk topological invariants.

Contribution

It defines a new edge index via spectral flow around a boundary singularity, connecting boundary conditions with bulk Chern numbers in shallow-water models.

Findings

The edge index is stable and quantized due to topological spectral flow.

The anomaly in bulk-edge correspondence is resolved through spectral flow analysis.

The spectral flow structure of the model is comprehensively characterized.

Abstract

In the context of topological insulators, the shallow-water model was recently shown to exhibit an anomalous bulk-edge correspondence. For the model with a boundary, the parameter space involves both longitudinal momentum and boundary conditions, and exhibits a peculiar singularity. We resolve the anomaly in question by defining a new kind of edge index as the spectral flow around this singularity. Crucially, this edge index samples a whole family of boundary conditions, and we interpret it as a boundary-driven quantized pumping. Our edge index is stable due to the topological nature of spectral flow, and we prove its correspondence with the bulk Chern number index using scattering theory and a relative version of Levinson's theorem. The full spectral flow structure of the model is also investigated.