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In the presence of a random potential due to the presence of disorder, the degenerate Landau levels split into a band. It is given that the states in the middle of this band are extended and the ones at the high and low energy tails are localized. How is this conclusion reached?

Also, when the magnetic field is varied from the values at the middle of the plateau (i.e from the value for which $\nu$ Landau levels are fully filled), the electrons must now fill the localized states in the band so that there is still no current in the longitudinal direction, and $\rho_{xx}$ and $\rho_{xy}$ retain their values. However, since we also know that the bulk states cannot conduct and that the edge states are responsible for the conduction, does it mean that localized states at the tails of the band correspond to the bulk while the extended states in the middle of the band correspond to the edges?

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To answer on your first question. So we add some random potential $V(x)$ which leads to the broadening of the Landau levels forming some kind of band states, having some Lorentzian shape, which are centered at the unperturbed energy of each Landau level. Now it is clear that states close to the centre/middle of this band, close to the unperturbed energy will be less affected by the random potential than those states further away from the unperturbed energy. Hence, these latter states are much more affected by the random potential and as a consequence they become localized, whereas the states close to the middle remain extended states.

To answer your second question, the localized states are indeed representing the bulk of the material, whereas the extended states will become edge states. Often they consider these edge states as some kind of skipping orbits generating an edge current.

I should recommend you reading the lecture notes of Prof. Tong on 'Quantum Hall Effect': http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf

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