Classifying Bulk-Edge Anomalies in the Dirac Hamiltonian

Abstract

We study the Dirac Hamiltonian in dimension two with a mass term and a large momentum regularization and show that bulk-edge correspondence fails. Despite a well-defined bulk topological index (the Chern number), the number of edge modes depends on the boundary condition. The origin of this anomaly is rooted in the unbounded nature of the spectrum. It is detected with Levinson’s theorem from scattering theory and quantified via an anomalous winding number at infinite energy, dubbed ghost charge. First, we classify, up to equivalence, all self-adjoint boundary conditions, using Schubert cell decomposition of a Grassmannian. Then, we investigate which ones are anomalous. We expand the scattering amplitude near infinite energy, for which a dominant scale captures the asymptotic winding number. Remarkably, this can be achieved for every self-adjoint boundary condition, leading to an exhaustive anomaly classification. It shows that anomalies are ubiquitous and typical. Boundary conditions with a ghost charge of 2 are also revealed within the process.