I've been trying to understand IQHE by reading these lecture notes by David Tong.

Mainly, I was trying to understand the quantized hall resistivity in terms of the number of Landau levels crossing the fermi energy.

Then, I began thinking about why spin induced Zeeman splitting is never really mentioned in the context of IQHE.

The lecture notes say that it's because typically the Zeeman splitting is very small and it polarizes the spin of the electron.

I think the spin based splitting of energy states still confuses me because in my mind, with the spins of electrons taken into account, you have twice as many energy states crossing the fermi energy.

The filling factor in IQHE is the number of landau levels crossing the fermi energy (as shown in the image below). To me, the spin Zeeman splitting seems to double that number.

enter image description here

  • $\begingroup$ It looks like you read the notes backwards. They say the Zeeman splitting is large, not small. $\endgroup$
    – knzhou
    Nov 17 '18 at 9:47
  • $\begingroup$ $\uparrow$ Which page? $\endgroup$
    – Qmechanic
    Nov 17 '18 at 10:00

This doubling indeed happens in the "quantum spin Hall effect", but those systems are at zero magnetic field (and moreover enjoy time reversal symmetry, which is key). However, in the usual quantum Hall effect, there is a huge static magnetic field, which polarizes all the low energy electrons (the Zeeman splitting is large because it goes like $|B|$). Therefore, they are essentially spinless. See this review http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf , section 1.4.


The definition of filling factor is, $\nu \equiv \frac{\text{number of particles}}{\text {number of flux quanta}}$. I guess even if you include the Zeeman splitting this is not going to change.


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