I think that I may have an answer to my question here.

Perhaps, the right question to ask is not whether an "ordinary" quantum Hall system (QH) in 3D and quantum anomalous Hall system (QAH) is connected through some adiabatic path. When the 3D TKNN invariant is valid, we see that Hall conductance is independent of the external magnetic field:

\begin{equation}
\lim_{\vec{B}\to0} \sigma_{ij} = \lim_{\vec{B}\to0}\epsilon_{ijk}\frac{\partial \rho}{\partial B_k} = \frac{e^2}{2\pi h} \epsilon_{ijk} G_k,
\end{equation}

for some reciprocal lattice vector $\vec{G}$ from Ref. 23 in the question. Then, perhaps, all we can conclude QAH is a special case of 3D QH.

In general, QAH may be formed by a variety of physics (such as ferromagnetism and others as discussed here: http://10.1103/RevModPhys.83.1057), QAH may not necessarily be related to QH, and QAH may not need to be a limit of QH at all.

Moreover, Streda's formula may be regarded more as a theoretic "shortcut" to Hall conductance. Experimentally, the distinction between QAH and QH is that chiral modes exist for QAH at zero magnetic fields, which can be observed by supplying an external electric field. The bulk charge density $\rho$ is not meaningful in the sense that it does not couple to experimental probes.

This post imported from StackExchange Physics at 2020-09-11 18:14 (UTC), posted by SE-user Congmiao