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

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.

Unveiling topological modes on curved surfaces

“…The (generalized) Chern number given by this formula is related by a homotopy, with respect to some parameter, to the Chern number of the Hermitian system [37]. Also, it is important to notice that the shallow water equations, in fact, do not possess well-defined topological numbers in the flat case [39]. As was shown in Ref.…”