Anomalous bulk-edge correspondence in continuous media

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 cutoff. 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, which have a signature at finite frequency both in the local density of states and in a channel geometry. We provide a general formalism based on scattering theory to detect all edge modes properly, so that the bulk-edge correspondence is restored in a broader sense, implying in particular that chiral edge modes are not necessarily topological, and conversely. Our approach is illustrated through the odd-viscous shallow-water model and the massive Dirac Hamiltonian.