I am reading a review article on Weyl semimetal by Burkov where he writes, top of page 5:
A 3D quantum anomalous Hall insulator may be obtained by making a stack of 2D quantum Hall insulators [Ref. 23].
Ref. 23 in his paper is the generalization of the 2D TKNN invariant to the 3D case.
I am a little confused about what Burkov meant here. The TKNN invariant in 3D was derived in the presence of a nonzero external magnetic field. I understand that, from Haldane's model of a Chern insulator, with Streda's formula, the quantum anomalous Hall conductance is the limit (for a 2D system):
\begin{equation} \lim_{B_k\to 0} \sigma_{ij} = \lim_{B_k\to 0} \epsilon_{ijk} \frac{\partial \rho}{\partial B_k} \neq 0, \end{equation}
where $\rho$ is the electric-charge density, $B_k$ is the external magnetic field, and $\{i,j,k\}$ are spatial indices. So, it seems that Burkov is implying that the above limit exists in 3D. If such a nontrivial limit indeed exists in 3D (which I think is true after reading Sec. III of Ref. 23), doesn't this imply all time-reversal broken quantum Hall systems in 3D are also quantum anomalous Hall systems? This seems a little odd to me. What is then the difference between an "ordinary" quantum Hall system and a quantum anomalous Hall system? Do they belong to the same topological phase (i.e. connected by a continuous adiabatic transformation)?