Generalized topological bulk-edge correspondence in bulk-Hermitian continuous systems with non-Hermitian boundary conditions

TL;DR

This paper extends the bulk-edge correspondence to non-Hermitian boundary conditions in continuous topological systems, showing that edge modes are determined by roots of the scattering matrix and maintaining the topological structure with modified analysis.

Contribution

It introduces a generalized bulk-edge correspondence for non-Hermitian boundary conditions in continuous systems, revealing how edge modes relate to scattering matrix roots and modifying Levinson's theorem.

Findings

Edge modes emerge at roots of the scattering matrix, not poles.

The topological structure remains consistent with Hermitian cases under modified contours.

The generalized BEC applies to systems with odd viscosity and large scattering matrices.

Abstract

The bulk-edge correspondence (BEC) is the hallmark of topological systems. In continuous (nonlattice) Hermitian systems with an unbounded wave vector, it was recently shown that the BEC of Chern insulators is modified. How would it be further affected in non-Hermitian systems, experiencing loss and/or gain? In this work, we take the first step in this direction, by studying a bulk-Hermitian continuous system with non-Hermitian boundary conditions. We find in this case that edge modes emerge at the roots of the scattering matrix, as opposed to the Hermitian case, where they emerge at its poles (or, more accurately, coalescence of roots and poles). This entails a nontrivial modification to the relative Levinson's theorem. We then show that the topological structure remains the same as in the Hermitian case, and the generalized BEC holds, provided one employs appropriately modified contours in the wave-vector plane so that the scattering matrix phase winding counts the edge modes correctly. We exemplify all this using a paradigmatic model of waves in a shallow ocean or active systems in the presence of odd viscosity, as well as 2D electron gas with Hall viscosity. We use this opportunity to examine the case of large odd viscosity, where the scattering matrix becomes $2\times2$, which has not been discussed in previous works on the Hermitian generalized BEC.