What's the relation between quantized Hall effects and topology materials? The quantized Hall effects (ignoring fractional Hall effect) include:

*

*Quantum Hall effect;

*Quantum anomalous Hall effect;

*Quantum spin Hall effect.

All these effects are just describing the transport phenomena of materials or physical systems driven by electromagnetic fields. The driven force is clearly described below:

Interestingly, all these effects are related to topology. On the other hand, many topological materials (like topological insulators/semimetal, Dirac and Weyl semimetal ...) are discovered. I want to know there are any relations between these topological materials and the quantized Hall effects? For example, in this paper, the author says that the quantum spin Hall (QSH) states are topological insulators.
 A: The label "topological material" has been fairly liberally applied to a wide class of materials, and there is not always a clear agreement what should qualify as topological. However, there are some features which are common across most topological materials. Such features include:

*

*Unusual surface states

*Quantized response functions

*Bulk quantized invariants

Though of course this is not an exhaustive list, not all types of topology have all of these features and these features are interdependent.
The quantum hall effect (in its many forms) are related to the second point on that list: Quantized response functions.
In particular, the famous TKNN invariant relates the hall conductivity $\sigma_{xy}$ (a response function) directly to a quantized topological invariant (the Chern number) through:
$$
\sigma_{xy} = -\frac{e^2}{2\pi \hbar}\sum_{\alpha}C_{\alpha}
$$
Where the sum is over the filled bands and $C_{\alpha}$ is the Chern number of band $\alpha$.
The topological invariant $C_{\alpha}$ is quantized to be an integer, and therefore the response function is quantized (in units of $e^2/{2\pi \hbar}$).
The interplay of response functions and topology is profound, as usually nontrivial topology is detected through some response-type experiment. There are many other examples, including the quantized circular photogalvanic effects (an optical response) associated with certain Weyl semimetals or the quantized magnetoelectric responses, associted with certain magnetic TIs
A good review of the interplay of response functions and topology can (in my opinion) be found in D. Vanderbilt's Berry Phase in electronic structure theory (2018), though most texts on topological insulators should cover this link.
