Topology of 2D Dirac operators with variable mass and an application to shallow-water waves

TL;DR

This paper addresses topological issues in 2D Dirac operators with variable mass and shallow-water models by introducing spatial variation to define proper indices, thereby restoring bulk-edge correspondence.

Contribution

It introduces a method to define proper bulk and edge indices for 2D Dirac operators with spatially varying mass, resolving topological inconsistencies.

Findings

Proper bulk and edge indices can be defined for variable-mass Dirac operators.

The bulk-edge correspondence is restored through the introduced approach.

The method applies to shallow-water wave models with similar topological issues.

Abstract

A Dirac operator on the plane with constant (positive) mass is a Chern insulator, sitting in class D of the Kitaev table. Despite its simplicity, this system is topologically ill-behaved: the non-compact Brillouin zone prevents definition of a bulk invariant, and naively placing the model on a manifold with boundary results in violations of the bulk-edge correspondence (BEC). We overcome both issues by letting the mass spatially vary in the vertical direction, interpolating between the original model and its negative-mass counterpart. Proper bulk and edge indices can now be defined. They are shown to coincide, thereby embodying BEC. The shallow-water model exhibits the same illnesses as the 2D massive Dirac. Identical problems suggest identical solutions, and indeed extending the approach above to this setting yields proper indices and another instance of BEC.