Axion coupling in the hybrid Wannier representation

One of the most important quantized responses of 3D topological insulators (TIs) is due to the axion coupling θ. Strong TIs, axion insulators, and the topological magnetoelectric effect owe their robustness to the quantization of the axion coupling (the axion Z₂ index). Time reversal was the first quantizing symmetry to be recognized, but recently a plethora of magnetic symmetries were found to have the same effect. After we enumerate all the quantizing symmetries, we explore how the nature of these symmetries affects the topological properties of the system. One tool that has proven especially useful in that respect is the hybrid Wannier (HW) representation. In this representation the quantizing symmetries can be divided into three distinct classes. By analyzing each of them in turn, we are able to explain whether the connectedness of the HW band structure in a topological phase is robust or fragile. Furthermore, we clarify the correspondence between the connectedness of the HW band structure and the appearance of either topologically protected metallic surfaces or insulating surfaces with half-quantized anomalous Hall response. We corroborate our results with various tight-binding models.