Orbital Magnetoelectric Effect in Three-dimension Chern Insulators

Orbital magnetoelectric effect is closely related to the band topology of bulk crystalline insulators. Typical examples include the half quantized Chern-Simons orbital magnetoelectric coupling in three dimensional (3D) axion insulators and topological insulators, which are the hallmarks of their nontrivial bulk band topology. However, the Chern-Simons coupling is well defined only for insulators with zero Chern number, the orbital magnetoelectric effects in 3D insulators with nonzero (layer) Chern numbers are still open questions. In this work, adopting the definition of layer-projected orbital magnetization, we derive a new quantization rule in the layer-resolved orbital magnetoelectric response in 3D Chern insulators, which is quantized in units of the layer number multiplied by $e^2/h$ and is protected by lattice translational symmetry. In order to verify the quantization rule derived from theoretical analysis, a model Hamiltonian of quasi-3D Chern insulator is constructed based on the Haldane model, and the response of the layer-resolved orbital magnetization to the applied electric field is numerically calculated. Our calculations indicate that the quantized orbital magnetoelectric response remains robust for various types of interlayer hoppings and stackings. Disorder and susbtrate effects are also taken into account to test the robustness of the quantized response.