Cross-posted and answered at Physics.SE.

The quantized versions of the Hall effect include (ignoring fractional quantum Hall effect):

  • Quantum Hall effect;
  • Quantum anomalous Hall effect;
  • Quantum spin Hall effect.

All these effects are transport phenomena but driven by different mechanisms, illustrated below: enter image description here

On the other hand, the quantized Hall conductivity is closely related to Berry curvature, which can be viewed as the magnetic field in reciprocal space, as justified by the semiclassical equation (Eqs.(5.8-a,b) in this paper) of motions for Bloch electrons: \begin{align} \hbar \dot{\vec{k}} & = -e \vec{E}-e\dot{\vec{r}} \times \vec{B} \\ \dot{\vec{r}} &=\dfrac{\partial \varepsilon_M(\vec{k})}{\hbar\partial\vec{k}}-\dot{\vec{k}} \times \vec{\Omega} \end{align} in which $\vec{\Omega}$ is the Berry curvature. Because the $\vec{\Omega}$ is explicitly incorporated into the equations of motion for Bloch electrons, so it must be playing some role like the magnetic field $\vec{B}$ in the Hall effect. So what's the role of Berry curvature in various quantized Hall effects?

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    $\begingroup$ I just figured out that this same question was also asked on Physics.SE. If you cross-post a question, especially if the other site gets an answer first, then it's very helpful if you could let us know! It prevents duplication of effort. Some sites take this even more seriously than we do, and the diamond moderators on those sites have been known to act more harshly on cross-posted questions. $\endgroup$ Jan 3 at 7:13