In the past few days I've become increasingly intrigued by the QHE, mainly thanks to very interesting questions and answers that have appeared here. Unfortunately, I am as of yet very confused by all the (seemingly disparate) stuff I learned.
First, here are some random points that I've been able to gather
- I(nteger)QHE occurs due to the presence of Landau levels
- IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone
- IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force
- F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge
- FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff
- FQHE has something to do with hierarchy states
So, here are the questions
- Most importantly, do these points make sense? Please correct any mistakes I made and/or fill in other important observations
- How do explanations 1. and 2. of IQHE come together? Landau quantization only talks about electron states while topological picture doesn't mention them at all (they should be replaced by global topological states that are stable w.r.t. perturbations)
- How do explanations 4., 5. and 6. relate together
- Is there any accessible introductory literature into these matters?
- Do IQHE and FQHE have anything (besides last three letters) in common so that e.g. IQHE can be treated as a special case? My understanding (based on 3.) is that this is not the case but several points hint into opposite direction. That's also why I ask about both QHE in a single question
Best Answer
Oh boy, hard to know where to start. Let me begin and see where I run out of steam. I'll go by the order you wrote your questions and make comments:
When you quantize electrons in a magnetic field, you get Landau levels: discrete energy levels which are highly degenerate. You can visualize each one of them as an electron moving in a circle whose radius is quantized (determined by the Landau level) and whose center can be anywhere (resulting in the degeneracy). Contrary to some discussions you hear sometimes, this by itself does NOT result in quantized Hall conductance.
For the integer QHE, the next crucial step is the presence of a random potential, provided by impurities. Then one can show that each Landau level contributes a fixed value to the Hall conductance, and therefore that conductance counts the number of filled Landau levels. The fact that this is robust is related to the topology, the Chern number and all that good stuff.
FQHE is a different story, for which the Hall conductance can be fractional. The only thing IQHE and FQHE have in common is the ultimate physical effect, but the mechanism is very different. For the fractional effect you need very pure samples, since it is driven by strong Coulomb intercations between the degenerate electrons in each Landau levels. This is an inherently difficult problem, and in fact it was solved only by a guess - the Laughlin wavefunction.
The EFT that describes the low energy excitations is related to the Chern-Simons theory, and those basic excitations obey anyonic statistics. Beyond that, I think all other effects you mentioned (e.g. heirarchy states), could be described as "special topics".
Finally, I am just a humble high energy theorist, so I'll wait for corrections and more complete picture from the experts. Still, that was fun to write.
(Incidentally, all of this is well-known stuff appearing in textbooks, though not always in an organized way. One good source: Mike Stone has edited a collection of papers on the subject for which he provided a series of introductions. If you find this book, those introductions are very good.)