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.
REFERENCES
In the context of band theory, it is natural to restrict a domain of the definition of the Bloch wavenumber k to the first Brillouin zone, but, here, we consider the Bloch wavenumber k in the extended Brillouin zone19 for simplicity. In this manuscript, our main interest is a local classification of Hamiltonians in a neighborhood of band crossings and bifurcations of the band crossings. There, the difference between the two representations does not make any difference.
Here, we would like to consider the class of diffeomorphisms that do not necessarily fix the origin because if we apply a generic perturbation to the Hamiltonian having a band crossing at the origin, the band crossing typically moves away from the origin but the Hamiltonian in a neighborhood of the band crossing does not necessarily change qualitatively.39
This result follows from Ref. 29, Lemma A.9.2. in p. 484 because formal power series (for both vector and scalar fields) are equivalent to smooth functions modulo by the Borel-Ritt theorem.