# Laughlin's topological argument

I have a confusion about understanding the Laughlin's topological argument on Hall conductivity quantization.

This argument states that the Hall conductivity is $$\sigma_{xy}=\frac{e}{h}Q,$$ where $Q$ is a charge transferred from the left reservoir $A$ to the right reservoir $B$ when one quantum of magnetic flux $\Phi=\Phi_0$ is inserted through the cylinder. Then it is usually told that during the flux insertion some filled states become empty and some empty states become filled. Hence $Q=en$ with $n$ being an integer number of transferred electrons so $\sigma_{xy}=(e^2/h)n$ is quantized.

I don't understand: what exactly happens with extended and localized states during the flux insertion? As known, highly degenerate Landau levels break up in the presence of disorder into extended and localized states.

What happens when the magnetic flux is inserted? As known,

1) the localized states are not affected by the flux $\Phi$ since their wave functions don't enclose the cylinder and undergo only a gauge transformation;

2) the extended states change with $\Phi$ but return to their initial form when $\Phi=\Phi_0$;

3) since a disordered system is usually a chaotic one, its energy levels don't cross (energy level repulsion);

4) since the flux is inserted adiabatically slowly, all transitions between different single-particle levels are forbidden thus all filled states remain filled and all empty states remain empty;

5) the Fermi level is placed inside the mobility gap, where only localized states exist.

With taking into account these arguments, evolution of energy levels with $\Phi$ changing from $0$ to $\Phi_0$ should look like that in the picture below.

As seen, in this picture the fillings of single-particle states don't change. So how it can happen that some integer number of electrons get transferred from localized states of the left reservoir to other localized states of the right reservoir?

• Doesn't the DOS change under the addition of flux quanta, namely the lines get scrunched up closer to the origin? – Aaron Oct 5 '17 at 15:21
• @Aaron It is interesting scenario, when extended states move from one Landau level to another (diagonal motion of red lines in the picture), I thought about it. But, in this case they should cross localized energy levels (blue lines), which separate different Landau levels. Since the crossing is forbidden, we should have some anticrossings instead... – Alexey Sokolik Oct 5 '17 at 15:30
• Why is there no diagonal motion of localized states as well? – Aaron Oct 5 '17 at 16:23
• @Aaron Because localized states are not affected by the flux at all (except unobservable gauge transformation), they do not undergo the Aaronov-Bohm effect since their wave functions does not enclose the flux. The same refers to electron states in the both reservoirs, so I don't understand how one of them can be filled and other emptied. – Alexey Sokolik Oct 5 '17 at 17:20