Local models and global constraints for degeneracies and band crossings

We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus of possibly higher order band crossings aka. degeneracies, we add the study of local models. These give more detailed information including that of the local Chern classes and Berry phases. We then give global constraints for the topological invariants in a systematic fashion. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. We also discuss how the results can be used in the study of deformations. This theory of local models and global constraints is then applied to examples which include the Gyroid geometry, which exhibits two Weyl points and two triple crossings and the honeycomb geometry with its two Dirac points. (C) 2020 Elsevier B.V. All rights reserved.