May 4, 2010


I started to talk about the toric code some time ago (see Kitaev's Toric Code) and finished writing that errors in the codeword, which if you remember was constructed using spins on the surface of a torus, can be interpreted as pair of quasiparticles which behave like anyons. Although every physicist reading this knows what a quasiparticle is since their first Ashcroft and Mermin reading (the book linked to at the left, which I like a lot), I believe that for a broader audience this term is not entirely familiar.
So let me try to explain it before going to anyons. A quasiparticle is not supposed to be a "real" particle, at least in the sense that it is not an elementary particle or a bound state of elementary particles, although I know that the philosophers out there will pick on me because of this statement.

At the most fundamental level, nature is described by a quantum theory of fields, QFT for short. In this model of reality, the world is composed of fields and the elementary particles is what we detect when those fields are not in their ground state, which means, they have energy above their minimum energy levels. These excited states are described by the action of operators, mathematical functions acting on vectors, that create and anihilate a certain number of particles in the system when acting on a vector representing the lower energy state |0>, which we call "the vacuum" for obvious reasons (there are no particles).

The mathematical entity that embodies the features of the system is called the Lagrangean and can be written in a certain specific way using the creation and annihilation operators. In some situations, specially more complicated systems like condensed matter ones, there is a way to write the Lagrangean in a mathematical analogous form in terms of some sort of creation/annihilation operators. Although they don't appear because of "real" particles in the system, they have all the mathematical structure the real ones have. These operators create excited levels that are analogous to the particle states and that is how they end up being considered as (quasi-)particles. The term seems to have been coined by Lev Landau, one of the greatest physicists of the last century whose name didn't make its way to the world media.

I know what you must be thinking now... How do we know that our elementary particles are not quasiparticles in some medium, say, the aether. Well, some people have proposed that and there is an entire book written by a a very competent physicist (although I am not trying to make use of an authority fallacy here...) about the subject:

[1] The Universe in a Helium Droplet, G. Volovik

But let's not loose the focus here. Now it became clear why we call our errors quasiparticles. They are obviously not any kind of elementary particle travelling around in our system, but can be mathematically described as such. Let's move on to anyons. Now, think about the following process. Imagine two elementary particles, let's say, two electrons. Move one electron very slowly around the other until the former gets back to the initial position. If we do it very slowly indeed , adiabatically for quantum physicists or quasi-staticallyfor classical ones, in the end of the process the description of the system should simply comes back to the initial one (assuming that nothing else in the neighbouring universe has changed...). I am going to jump over a lot of things now. For those of you who will be annoyed by that, a more rigorous exposition can be found here:

[2] An Anyon Primer, S. Rao [arXiv:hep-th/9209066v3]

The main point is that it was understood some time ago that this result (nothing changing) is a consequence of a very beautiful geometrical property of three-dimensional space. If you visualise the path of the moving electron as a rubber band, you see that in 3D you can continuously shrink the band to a point without any impediment. But you would not be able to shrink the band if both electrons lived in a 2D plane. If you try, the band will always find the second electrons in its way and could not shrink. Because of this, in this case things may not be exactly as before.

For those particles for which winding around would not change the system, there are two possible kinds of behaviour defined for the statistics they obey: fermions, obeying Fermi-Dirac statistics, and bosons, obeying Bose-Einstein statistics. But those that do change do not need to obey only these two statistics, they can obey ANY statistics and then they are called anyons.

You see, the toric code is defined on a 2D surface, the ideal place for the appearing of anyons, and that's exactly how the error can be described mathematically. In the next post about this subject I will explain it in more details. Actually, I am stopping now because I am also learning this subject and I need to understand it better before writing it here. As such, if anyone can pick something wrong here, please correct me. And also, if you have anything to add or any question, please post it in the comments and we can learn it together.

Finally, let me link to some other Wikipedia articles that may be of relevance here:

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