Bulk-edge correspondence for unbounded Dirac-Landau operators

TL;DR

This paper establishes a bulk-edge correspondence for unbounded magnetic Dirac operators in two dimensions, linking spectral properties, topological invariants, and the behavior of edge states through advanced PDE and operator techniques.

Contribution

It introduces a novel topological framework connecting the spectral characteristics of unbounded Dirac operators with edge state phenomena using integral operator methods.

Findings

Derivation of a Středa formula for spectral islands

Identification of the Chern character as a key topological invariant

Establishment of a linear variation of the density of states with magnetic field

Abstract

We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying functions of the bulk and edge operators. One of these corollaries leads to the so-called St\v{r}eda formula: if the bulk operator has an isolated compact spectral island, then the integrated density of states of the corresponding bulk spectral projection varies linearly with the magnetic field as long as the gaps between the spectral island and the rest of the spectrum are not closed, and the slope of this variation is given by the Chern character of the projection. The same bulk Chern character is related to the number of edge states which appear in the gaps of the bulk operator.