174. Frank Wilczek on anyons
Save This Post as PDFAnother interesting recent paper in archives which a undergrad student will be able to read is the paper “New kinds of quantum statistics” by Frank Wilczek. I would say it is actually useful to read it irrespectively whether you are going to specialize in quantum field theory and string theory, condensed matter physics or quantum computing. As follows from the title of the paper, Wilczek is talking about quantum statistics different from Fermi-Dirac QM statistics (corresponding to fermions) or Bose-Einstein statistics (corresponding to bosons).
1. Wait, isn’t he a crackpot?! There are only fermions and bosons in quantum mechanics!
No, he is not. If you consider a 3d quantum mechanical system, then indeed Fermi-Dirac and Bose-Einstein are the only two quantum statistics that are allowed. If you exchange fermions in a two-particle state, then the overall wave function will change the sign:
,
For bosons, it will not change at all:
.
However, yet another possibility can be realized in 2d quantum mechanical systems – the statistics can actually range continuously between Fermi-Dirac and Bose-Einstein: one can have
(1)
with arbitrary complex angle 
(note that 
corresponds to Bose-Einstein statistics, while 
– to Fermi-Dirac statistics).
Degrees of freedom obeying the statistics (1) are called (abelian) anyons.
2. Abelian anyons
Apart from his seminal work on asymptotic freedom of QCD, which he got the Nobel Prize for, Frank Wilczek has made many other substantial contributions into theoretical physics. One of them was discovery of the fact that relevant degrees of freedom in the fractional Hall effect obey anyon statistics (more accurately, excitations in Laughlin 1/n states are abelian anyons with 
).
Another, simple, example of a theory featuring abelian anyons is given in the Frank Wilzcek’s paper. Let us consider a 
gauge theory with particles that carry a charge 
. The gauge group is spontaneously broken by a condensate of another field that carries a charge 
, where 
is integer. A gauge transformation 
that leaves condensate invariant, multiplies the wavefunction of the q-charged particle by 
, so, in principle, it may change it non-trivially.
The unbroken gauge group is 
, and if the theory is 2-dimensional, it supports vortices with flux quantized in units of
,
and composites carrying non-trivial charge and flux are generally abelian anyons. You can easily see where the 2-dimensionality is important – non-trivial topological solutions in gauge theory only exist in 2d.
3. Non-abelian anyons
If the initial unbroken gauge group is non-abelian and we break it in the same fashion as above, the anyons will be non-abelian. To my knowledge, such excitations were not yet observed in Nature, but there are very concrete theoretical examples of QM systems, where non-abelian anyons appear such as spin 1/2 Heisenberg-like model on a honeycomb lattice.
By the way, Alexei Kitaev, the author of the paper that I cite above is quite a personality and a role model – try to Google him 
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I recently came accross your blog and have been reading along. I thought I would leave my first comment. I dont know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often.
Kate
Dear Kate,
you are very welcome.
Dmitry.
Hi Dmitry
I don’t know anything in this field, I just want to lean something. If I understand correctly, the axions are some quasiparticles, they are not fundamental, right? In fractional Hall effect, what acts as the axion?
Keith
Dear Keith,
Yes, they are composite – like as in the example that I used in the post – you can visualize them as a particle with a vortex constantly attached to it. I think I’ll write a blog post about quantum Hall effect for you.
Cheers,
Dmitry.
P.S. The excitations are called anyons, not axions.
Hi Dmitry
Thanks. I look forward to it.
Keith
Hi Keith,
started to discuss the quantum Hall effect here.
Cheers,
Dmitry.