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.
Highlights
Tools from topology are central to our understanding of a variety of physical phenomena [1], from quantized vortices in superfluids [2,3,4], to defects in ordered media [5,6,7], or to the description vorticity knots in classical fluids [8], among other applications
Such a correspondence is a hallmark of topology in physics which states that, when there exists a topological number associated to an infinite and gapped system, topologically protected edge modes appear in a sample with a boundary and vice versa
It was first realized in quantum Hall effect that both bulk and edge pictures were associated to topological quantities [12,13,14], which coincide [10,11]
Summary
Tools from topology are central to our understanding of a variety of physical phenomena [1], from quantized vortices in superfluids [2,3,4], to defects in ordered media [5,6,7], or to the description vorticity knots in classical fluids [8], among other applications. In the shallow water model, we observe that the number of edge modes depends on the boundary condition, be it with oddviscous terms [53], or without it [55] This looks suspicious compared to the expected topological nature of these modes in the presence of odd viscosity, and raises the apparent paradox of a violation of the bulk-edge correspondence. This anomaly is not restricted to the shallow-water model and was already noticed in other two-dimensional continuous models, e.g., in the valley quantum Hall effect [56] or compressible stratified fluids [57], that are both effectively well described by a Dirac Hamiltonian.
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