The transverse conductivity $\sigma_{xy}$ at zero frequency is quantized when the chemical potential $\mu$ is within the gap for a topological system. A typical plot of $\sigma_{xy}(\mu)$ is like the following. 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 $\sigma_H=\sigma_{xy}$. Let's ignore the difference here.
It is known that $\mathrm{Re}\,\sigma_H(\omega,\mu)$ for general $\omega$ will change sign at some $\mu$, where the type of main carrier changes from electron to hole or the like. An example is the following plot (solid line for real part) at some fixed $\omega$ from this PRL paper. It uses the Kubo formula for a particle-hole symmetric BdG system.
My question:
Does the sign change argument always hold at any frequency? I naively think so. It is used in experiments to determine carrier type, probably very often in dc transport. If it holds also for $\omega=0$, the first figure with quantized result seems to be contradictory?
