# Quantum Hall effect for dummies

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

1. I(nteger)QHE occurs due to the presence of Landau levels
2. 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
3. IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force
4. 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
5. FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff
6. 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
• Buy a copy of Jain's "Composite Fermions" and seal yourself in a comfortable room with plenty of snacks. You will emerge enlightened. – user346 Mar 1 '11 at 11:01
• Whilst I respect Jain's works, it is worthwhile pointing out that his books is obviously a biased view of the problem, and does not necessarily reflect a consensus of the community! Nevertheless, the composite fermions picture is nice in its intuitiveness and helps to build a mental picture. Incidentally, it is worth pointing out that some of the recent literature on topological insulators actually contain some of the cleanest expositions of the IQHE. – genneth Mar 1 '11 at 15:08
• @genneth I think you might be referring to a controversy over the "composite fermion" theory. Dr. Jain addresses this issue in his book actually. Despite Jain's obvious bias towards promoting his own perspective, I think this book remains the best introduction to the physics of the quantum hall effect. – user346 Mar 1 '11 at 19:50

Here are some comments on the points:

1) I(nteger)QHE occurs due to the presence of Landau levels

Yes

2) 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 is an example of topological order, although topological order is introduced to mainly describe FQHE. The characterization of IQHE by Chern number of energy band only works for non-interacting fermion with no impurity, while IQHE exists even for interacting fermions. So IQHE is more than the Chern number of energy band. The quasiparticles excitations in IQH states are always fermions.

3) IQHE requires negligible electron-electron interactions and so is dependent on the presence of impurities that shield from Coulomb force.

IQHE does not require negligible electron-electron interactions. IQHE exist even in the clean system with Coulomb force, if you control the electron density by gates.

4) 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 occures not because formation of anyons. Fermion alway carry Fermi statistics by definition, and they are never anyons. FQHE occures because of strong interacting effects. The effective non-interacting description does not really work (for example, it fails to describe the edge states and non-Abelian states).

5) FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff

FQH states contain a new kind of order: topological order. The low energy effective theories of FQH states are TQFTs (such as Chern-Simons theories). The quasiparticles excitations in FQH states are anyons.

6) Hierarchy states are examples of FQH states.

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:

1. 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.

2. 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.

3. 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.

4. 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.)

• Thanks a lot! I'll look at that intro and (hopefully) ask somewhat more focused questions later. But right now I just didn't know where to start as the topic of QHE seems quite huge. – Marek Mar 1 '11 at 9:58
• This answer is very, very good! – DaniH Mar 24 '12 at 15:26

This was too long to fit into a comment, so an answer it will have to be. This is all in supplement to @Moshe R.'s answer, which is excellent.

First, just to correct your statements (in addition to Moshe's): 3. Impurities do not screen anything. The electrons themselves provide the screening to make an independent electron approximation semi-justified (this is the usual Landau Fermi-liquid argument). Impurities however provide the basic scattering potential to achieve some Anderson localisation, which is crucial for actually getting the plateaus --- otherwise one would never get any resistance at all! Incidentally, understanding this point is crucial for understanding why the longitudinal conductance displays the spikes that it does.

More generally:

In condense matter, we don't get to have exact theories --- everything is a simplified approximation. As such, one will come across in the literature many different theories, which emphasise different aspects of the phenomenon, and have differing amounts of complexity and quantitative accuracy. At this point, it is fair to say that IQHE is well understood, the prevailing theory being a combination of topological states, impurity effects, and 2-parameter scaling theory (of both longitudinal and transverse conductances, ala Khmelnitskii). However, the theory of FQHE has not reached quite the same consensus. In some respects, FQHE is like a IQHE of electrons with extra flux "bound" to them (through an effective interaction due to Coulomb repulsion); in this picture, all the messiness (impurities), etc. are again crucial. This is also related to the hierarchical states because one can imagine binding more flux to the anyonic excitations and getting more IQHE states of those. However, it is clear that since the basic ingredient is the strong Coulomb interaction, without a systematic (the above is very much ad hoc) treatment it is impossible to be confident about the range of validity of the theory. Work on this aspect is on going (but to be fair, somewhat stalled --- it is sufficiently hard theoretically speaking that one is really looking for some fundamental break through in mathematics to finish it off).

• Thank you. Could you elaborate (or just give a reference) a little on the scaling theory and Khmelnitskii? I am not familiar with either. – Marek Mar 1 '11 at 13:10
• @Marek: my knowledge comes from my supervisor, and I suspect it is a little folklore-ish in nature. Khmelnitskii's work is a little hard to find in English, and mostly exist in JETP. References I've seen (but not read): Muzykanskii and Khmelnitskii, JETP Lett. 62, 76 (1995), and Khmelnitskii, JETP Lett. 38, 552 (1985). An English reference is Pruisken, Nucl. Phys. B 235, 277 (1984). The modern work tends to proceed via a field theory or replica theory model of disorder, and derive an effective non-linear $\sigma$-model for the diffusive transport, and from there find a scaling theory. – genneth Mar 1 '11 at 15:05
• @Moshe R.: Notice that FQHE is not IQHE of anyons --- the anyons only appear as the excitations. It is formal --- the idea is to justify that such a picture makes sense and predicts the right (say) excitations, but there's no "derivation" to be had to get it. In condensed matter this is not always a problem --- many things are really just guesses which work exceptionally well. The higher cleaniness is just a result of the composite IQHE being a bit more fragile; notice that for Anderson localisation to occur in 2D, one just needs a sufficiently large sample with arbitrarily small impurities. – genneth Mar 1 '11 at 16:35
• The key problem with current FQHE theories is the lack of a detailed quantitative theory of how the interaction brings about the new order --- one usually simply posits the state and show that it is gapped, i.e. safe from small disturbances. In practise, one could level the same criticism at IQHE, which relies on Fermi liquid arguments, which are also foundationally not really rigorous. Nevertheless, most people are far happier to accept that interactions may be neglected entirely, than somehow incorporating part of the interaction into a topological order, and neglecting the rest. – genneth Mar 1 '11 at 16:38
• @4tnemele: Fermi liquid theory has a semi-controlled expansion (viz. Shankar) in terms of renormalisation about the Fermi surface. However, my point is that for FQHE we have even less. For instance, what you describe (i.e. CS flux attachment) is what I meant by "partially" accounting for the Coulomb interaction; it is not a controlled expansion in any sense. I am somewhat (ahem) familiar with Altland & Simons; yes, I think it's there, but in no great detail (at least in the 1st edition). – genneth Mar 2 '11 at 10:22

There is a book that covers exactly the questions you asked:

If you are short of time (or money) - the book is based on his thesis: