Longitudinal resistivity in Integer Quantum Hall Effect

[explainplease]

I have studied the integer quantum hall effect mainly from David Tong's notes and i understand how the $\rho_{xy}$ is quantized in terms of the chern number. What I don't understand is
- how the chern numbers relate to the number of filled Landau levels though.
- I also don't understand the behaviour of $\rho_{xx}$ and why it spikes each time $\rho_{xy}$ jumps.

 

[radium]

Usually we consider the Hall conductivity when we are talking about the QHE as a topological phase. If you calculate the Hall conductance with the Kubo formula, you will see that it is an integral of the Berry curvature over the filled energy levels. The Landau levels are just the spectrum of energies, so the filled Landau levels are just like filled energy bands if you were to consider some crystal structure for example.

The xx resistance spikes because the system becomes a metal during the transition, i.e. the gap closes. That's why the Hall conductance is topological, you cannot change it without going through a phase transition where the band gap closes and the system becomes a metal.

One thing to note though is that the xx resistance is zero when the Hall conductance is quantized and the system is an insulator because the current is not flowing in that direction.

 

[explainplease]

Usually we consider the Hall conductivity when we are talking about the QHE as a topological phase. If you calculate the Hall conductance with the Kubo formula, you will see that it is an integral of the Berry curvature over the filled energy levels. The Landau levels are just the spectrum of energies, so the filled Landau levels are just like filled energy bands if you were to consider some crystal structure for example.
.

Thanks a lot for your reply. I understand the "landau levels being the filled energy bands" part, it's just that no correlation between the Chern number and the Landau levels that are filled was discussed, and reading through that, i would infer that each landau level (/filled band) contributes corresponds to a chern number that is one, and if we have 3 filled Landau levels i'd have C=3 and so on. I am not sure of that though, i am just trying to get a relation between both because there obviously exists one.

The xx resistance spikes because the system becomes a metal during the transition, i.e. the gap closes. That's why the Hall conductance is topological, you cannot change it without going through a phase transition where the band gap closes and the system becomes a metal. .

Alright! I see, thanks for explaining that, can you expand of the point that the Hall conductance is topological because i can't change it without going through a phase transition? Is that what it means to be topological? what I understood that it just stays the same no matter what geometry I have, or what material.

 

[radium]

Yes, the Hall conductance \sigma_{xy}=\nu \frac{e^{2}}{h} where \nu is the filling factor. In the fractional quantum Hall effect \nu can be fractional because of strong electron interactions which means the Landau level is not completely filled. The famous Laughlin state is \nu=1/3.

The topology in the quantum Hall effect is in the wave function. The winding number comes from a quantization condition required to make the wavefunction well defined when you adiabatically travel around a torus and pick up a Berry phase. The Berry phase is like a magnetic field in momentum space, so the quantization in the integer quantum Hall effect arises in a way very similar to the Aharonov Bohm effect and the Dirac monopole.