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 Středa 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 that appear in the gaps of the bulk operator.
Bibliografisk noteFunding Information:
H.C. acknowledges the partial support from Danmarks Frie Forskningsfond Grant No. 8021-00084B. The work of M.M. is supported by a fellowship of the Alexander von Humboldt Foundation. K.S. is especially grateful to S. Teufel for extending an invitation to visit the University of Tübingen during the spring semester of 2022 and to “Augustinus Fonden,” “Knud Højgaards Fond,” and “William Demant Fonden” for financially supporting the visit.
© 2023 Author(s).