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.

10+9 pages (main body + appendices); 4 figures

References in corpus (5)

• Classification of topological insulators and superconductors in three spatial dimensions
• Dirac materials
• Equality of the bulk and edge Hall conductances in a mobility gap
• Spontaneous edge currents for the Dirac equation in two space dimensions
• Asymmetric transport computations in Dirac models of topological insulators

Cited by in corpus (6)

• Classifying bulk-edge anomalies in the Dirac Hamiltonian
• Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
• Topological edge states of continuous Hamiltonians
• Symmetry-Driven Bulk-Edge Correspondence in Electron Magnetofluids at Finite Temperature
• Analytic bulk-edge connection in circular-symmetric models
• Topological junctions for one-dimensional systems