The bulk-edge correspondence is a cornerstone theorem in condensed matter physics, linking a topological index of an infinitely extended system (the bulk) to the number of chiral edge modes at the boundary of a half-infinite sample (the edge). Meanwhile, the massive Dirac Hamiltonian serves as a quintessential model for topological insulators, widely employed as a low-energy effective theory. Surprisingly, this correspondence breaks down for the Dirac Hamiltonian: even with a properly regularized bulk, the number of edge modes depends on the choice of boundary conditions.
This talk aims to uncover the origins of this anomaly, which is tied to the presence of a "ghost" topological charge in the asymptotic part of the spectrum. We will present a generalized bulk-edge correspondence theorem for unbounded operators, employing scattering theory and a relative form of Levinson's theorem. Additionally, we will provide a complete classification of anomalous boundary conditions, revealing that such anomalies are not rare but ubiquitous and typical.
