# Why bulk states in quantum hall effect do not contribute to electric conductivity

Most reviews and textbooks explain quantum hall effect as insulating bulk states and conducting edge states, as is shown in the following picture. My question is: why bulk states are insulating in quantum hall effect(if one or more landau levels are filled)?

Here is one more consideration:

Some literature says impurity is necessary for insulating bulk states. However, laughlin's gauge argument(http://journals.aps.org/prb/abstract/10.1103/PhysRevB.23.5632) does not contain anything about disorder or impurity. Chern number, which characterises hall response to magnetic field, says nothing about disorder or impurity. Is impurity really necessary to observe QHE?

The tricky part is to explain the plateaus in $\sigma_{xy}$ and why they coincide with vanishings of $\sigma_{xx}$. To make this more evident let us suppose first you had a system without disorder, say you have an interacting Galilean that is fully translationally invariant. Ideally this would be realized e.g. in a Torus. Then you can show, following Kohn's theorem, that the Hall conductivity is always exactly $\sigma_{xy}=\nu e^2/h$, and that $\sigma_{xx}=0$. This is completely independent of the microscopic state of the system, e.g. it woud be true if you had a Laughlin state or a Charge density wave or anything. It follows from the fact that the center of mass of the system as a whole moves as a free charged particle in electric and magnetic fields. Thus there are no plateaus in this translationally invariant limit: the Hall conductivity traces a straight line when you plot it vs $\nu$.
Therefore, what is non-trivial is to explain why the Hall conductivity gets "stuck" at $\sigma_{xy}=\nu_0 e^2/h$ when you change the filling $\nu$ slightly away from a special filling $\nu_0$, namely, the existence of a "plateau" and its coincidence with the vanishing of $\sigma_{xx}$. Essentially, the picture is that as you change the electron density (filling) away from $\nu_0$, they are added into states localized by the disorder and don't contribute to transport (A caveat is that there can be bulk states deep below the chemical potential that are delocalized and can contribute to current flow, see previous reponse).