228. Book review: D. Yoshioka. The quantum Hall effect
Save This Post as PDFAbout a week ago we have briefly discussed the physics of the quantum Hall effect. Let me remind you the conclusions we came to.
We discriminate between the integer and fractional quantum Hall effects according to the value of the filling factor 
. The latter characterizes the many-particle wave function describing behaviour of electrons in the sample. effectively depends only on which Landau levels are filled and which – aren’t, i.e., on 
and ultimately on Landau levels themselves (that is, the value of magnetic field we apply to the sample).
Integer quantum Hall effect is the situation when is integer. It is fairly understood by using approximation of the weakly interacting electron gas in external magnetic field.
Fractional quantum Hall effect corresponds to the situation when is fractional 
The approximation of weakly interacting electron gas turns out to be inapplicable. Nevertheless, it seems to be possible to explicitly write down the many-body (multiparticle) wave function describing behavior of electrons in the sample, and one can somewhat understand its structure by visualizing collective excitations in the electronic liquid as electrons with quanta of magnetic flux attached to them (these excitations were denoted as anyons).
The logic of composite fermions proves to be possible to explain almost all FQHE states except the ones corresponding to filling factors with even denominator. The state 
is of especial interest for a theorist – as it seems, it corresponds to non-abelian anyons.
Although this is pretty much all we currently know about quantum Hall effect in a nutshell, my discussion of it wasn’t terribly deep. If you are a graduate student eager to explain the nature of mysterious FQHE state, reading my blog posts won’t probably be enough for you to start working on the problem.
What will be enough? As you may imagine, enough will be if you read a nice textbook on the subject. Some weeks ago, I have put a hand on such a textbook and wanted to share my experience with you. The book I am talking about is the one by Daijiro Yoshioka, called “The quantum Hall effect“.
The book is simply great and really stands separately from other text books on the quantum Hall effect. Why do I think so? First of all, it contains everything you need to know about QHE. Second, it is good for a student – because the author cared to add exercises after every Section, and there is no better way to get some understanding of the subject than solving a couple of nice related problems. Last but not least – it is short (!), that is – readable. You’ll need a week/couple of them to go through it (especially if you already know quantum mechanics), and you are on the train. Even if you decide to postpone solving puzzle , you can be sure that you’ll know enough to talk to experts about quantum Hall effect and don’t seem to be ignorant outsider to them
Let me go quickly through the book. Yoshioka starts explaining what are the systems where and what are the conditions under which the quantum Hall effect effect can be observed (as well as how it was originally discovered).
Then, he explains all relevant quantum mechanics you need to know (and even has a short introduction into Anderson localization which is a fascinating topic by itself).
Of course, he explains integer quantum Hall effect in details, but discussion of the fractional Hall effect is where the book really shines (he extensively discusses the Laughlin’s variational method – one of the best and clearest explanations why it works – and composite fermions).
Finally, he even talks about 
and states – not bad for the basic level textbook on the quantum Hall effect!
Well, anyway, to conclude – I’ve read it and have found that it was a great investment of time. It also gave me a lot of fun.
Cheers.
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I consider Yoshioka’s book to be nearly the worst of the the first tier quantum Hall books. There is a 2nd tier of books that should never be mentioned
The best book, however, and most comprehensive would be Jainendra Jain’s new book “Composite Fermions”. It provides the most comprehensive description of the integer and fractional Hall effects, describes composite fermions, and discusses most new topics including 5/2 and non-abelian anyons.