Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
Gian Michele Graf, Alessandro Tarantola
TL;DR
This work probes the limits of bulk-edge correspondence (BEC) for the rotating shallow water model with odd viscosity by classifying all local self-adjoint boundary conditions on a half-plane. It builds a scattering-theoretic edge index framework, connecting edge-state mergers to a bulk Chern number $C_+=2$ via a winding of the scattering amplitude $S$, and introduces four boundary-condition families (DD, ND, DN, NN) with particle-hole-symmetric subcases. The main result is that BEC is typical only in the DD family, while violations occur for ND, DN, NN across extensive parameter regions; a unified mechanism ties these violations to spectral events and transitions in boundary data. The paper also delivers explicit recipes to compute the four integers $(P,I,E,B)$ from a given boundary condition and links them to the topology of the edge problem, with Levinson-type theorems clarifying the bulk-edge relationship. This advances the understanding of how boundary details govern topological transport in continuum hydrodynamic systems.
Abstract
Bulk-edge correspondence is a wide-ranging principle that applies to topological matter, as well as a precise result established in a large and growing number of cases. According to the principle, the distinctive topological properties of matter, thought of as extending indefinitely in space, are equivalently reflected in the excitations running along its boundary, when one is present. Indices encode those properties, and their values, when differing, are witness to a violation of that correspondence. We address such violations, as they are encountered in a hydrodynamic context. The model concerns a shallow layer of fluid in a rotating frame and provides a local description of waves propagating either across the oceans or along a coastline; it becomes topological when suitably modified at short distances. The edge index is sensitive to boundary conditions, as exemplified in earlier work, hence exhibiting a violation. Here we present classification of all (local, self-adjoint) boundary conditions and a parameterization of their manifold. They come in four families, distinguished in part by the degree of their underlying differential operators. Essentially, that degree counts the degrees of freedom of the hydrodynamic field that are constrained at the boundary by way of their normal derivatives. Generally, both the correspondence and its violation are typical. Within families though, the maximally possible amount of violation is increasing with its degree. Several indices of interest are charted for all boundary conditions. A single spectral mechanism for the onset of violations is furthermore identified. The role of a symmetry is investigated.