The role of self-adjoint extensions in the bulk-edge correspondence

Abstract: We investigate the role of self-adjoint extensions in the bulk-edge correspondence for topological insulators. While the correspondence is well understood in discrete models with spectral gaps, complications arise in the presence of unbounded Hamiltonians and varying boundary conditions, leading to anomalous behavior that has recently been dubbed violations of bulk-edge correspondence. In this work we use a K-theoretic framework to identify precise conditions needed for unbounded Hamiltonians to be affiliated to the respective observable algebras and define K-theory classes. In special cases we can then exclude anomalous behaviour and obtain the standard bulk-edge correspondence, or, under weaker conditions, obtain a relative bulk-edge correspondence theorem, which compares pairs of Hamiltonians. Applying that relative approach in the bulk we recover among other things the so-called bulk-difference-interface correspondence for Hamiltonians that fail to define a bulk K-theory class in the conventional way. The second main result is that one can define K-theory classes in terms of von Neumann unitaries, which under changes in boundary conditions directly contribute to the number of protected edge states. This approach clarifies apparent violations of the classical bulk-edge paradigm and provides a systematic account of boundary-induced topological corrections.