The transverse conductivity $\sigma_{xy}$ at zero frequency is quantized when the chemical potential $\mu$ is within the gap for a topological system, such as quantum Hall effect and Haldane model. Another closely related quantity is the (optical) Hall conductivity $\sigma_H(\omega)=[\sigma_{xy}(\omega)-\sigma_{yx}(\omega)]/2$ for general $\omega$. In many cases, e.g., systems with $C_4$ rotation symmetry, we have $\sigma_H(\omega)=\sigma_{xy}(\omega)$. So Let's ignore their difference here.
This PRL paper calculates $\sigma_H(\omega)$ in the following figure (blue for the real part) at some $\mu$, which merely applied the standard Kubo formula to a BdG superconductor.
Why is the $\omega=0$ limit not quantized at all? Even if the system may not be topological, it can be just zero. My understanding is that $\mu$ is by construction in the gap in a BdG system. So if we write down its Hall conductivity, where is it different from the Chern number? Or put in other words, how is the link between Hall conductivity and Chern number broken in the present case which is not an insulator?
