Three-dimensional symmetry-breaking nontrivial topological states

We discuss topological electronic states described by the Dirac Hamiltonian plus an additional one in three dimensions. When the additional Hamiltonian is an element of an Abelian group, electronic states become topologically nontrivial even in the absence of the fundamental symmetries such as the time-reversal symmetry and the particle-hole one. Such symmetry-breaking topological states are characterized by the Chern number defined in the two-dimensional partial Brillouin zone. The topological insulators in Zeeman fields are an example of the symmetry-breaking topological electric state. We show the crossover from the topological insulating phase to the topological semimetal one in strong Zeeman fields.