Orbital magnetoelectric coupling of three dimensional 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. While the Chern-Simons coupling is well defined only for insulators with zero Chern number, the orbital magnetoelectric effects in 3D Chern insulators with nonzero (layer) Chern numbers are still open questions. In this work, we propose a never-mentioned quantization rule for the orbital magnetoelectric response in 3D Chern insulators, the spatial gradient of which is exactly quantized in unit of $e^2/h$. By theoretical analysis and numerical simulations, we demonstrate that such quantized orbital magnetoelectric response is exact for various types of interlayer hoppings and stackings, and remains robust even against disorder and lack of crystalline symmetries. We argue that the exact quantization has a topological origin and is protected by Chern number. Furthermore, we propose two promising material platforms to observe the proposed quantized orbital magnetoelectric response thanks to recent experimental developments in detecting spatial magnetic-field distributions in device systems.