Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities

Abstract

We classify two-by-two traceless Hamiltonians depending smoothly on a three-dimensional Bloch wavenumber and having a band crossing at the origin of the wavenumber space. Recently these Hamiltonians attract much interest among researchers in the condensed matter field since they are found to be effective Hamiltonians describing the band structure of the exotic materials such as Weyl semimetals. In this classification, we regard two such Hamiltonians as equivalent if there are appropriate special unitary transformation of degree 2 and diffeomorphism in the wavenumber space fixing the origin such that one of the Hamiltonians transforms to the other. Based on the equivalence relation, we obtain a complete list of classes up to codimension 7. For each Hamiltonian in the list, we calculate multiplicity and Chern number [D. J. Thouless et al., Phys. Rev. Lett. 49, 405 (1982); M. V. Berry, Proc. R. Soc. A 392, 45 (1983); and B. Simon, Phys. Rev. Lett. 51, 2167 (1983)], which are invariant under an arbitrary smooth deformation of the Hamiltonian. We also construct a universal unfolding for each Hamiltonian and demonstrate how they can be used for bifurcation analysis of band crossings.