Chern insulators in two and three dimensions: A global perspective

We introduce a second-quantized field theory for Chern insulators in both two and three dimensions, where the Hamiltonian features a static vector potential that has the periodicity of the crystal's lattice and leads to the spontaneous breaking of time-reversal symmetry in the system's ground state. This vector potential generates a magnetic field at the microscopic level that may be thought of as arising due to local moments associated with one or more magnetic ions in each unit cell. We write the Chern invariants characterizing the topology of the occupied “valence” bands, namely the Chern number in two dimensions and the Chern vector in three dimensions, in terms of the fundamental quantities in this Hamiltonian field theory, such as Bloch functions and velocity matrix elements, leading to “global” expressions that are well-defined across the Brillouin zone and involve the full band structure of these systems. Considering spinor electrons, we also discuss the symmetry properties of our Hamiltonian in the context of the three discrete symmetries of the tenfold way classification scheme, as well as inversion symmetry and gauge transformations. And we use this field theory to study the long-wavelength response of Chern insulators to electromagnetic fields in the optical regime, extending the usual quantum anomalous Hall response that occurs in the static limit to finite frequencies.