Orbital magnetoelectric coupling of three dimensional Chern insulators

Orbital magnetoelectric effect is closely related to the band topology of bulk crystalline insulators. While the half quantized Chern-Simons coupling is well defined only for insulators with zero Chern number, such as three dimensional (3D) axion insulators, the orbital magnetoelectric effects in 3D Chern insulators with nonzero (layer) Chern numbers remain unexplored. In this work, we propose a never-mentioned quantization rule for the layer-resolved orbital magnetoelectric response in quasi-3D and genuine 3D Chern insulators, showing that its spatial gradient is exactly quantized in unit of e²/h. We theoretically demonstrate that such quantized orbital magnetoelectric response, protected by Chern number, is exact for various types of interlayer hoppings and stackings, remaining robust even against disorder and lack of crystalline symmetries. Two promising material platforms for observing this effect are also proposed thanks to recent developments in detecting spatial magnetic-field distributions in device systems.