Anyons, Anyone?

A fundamental concept in modern physics is the idea of indistinguishable particles. All electrons, for example, as far as we can tell, are totally alike. All photons, as far as we can tell, are totally alike. Because these particles are indistinguishable, when we exchange such particles, this must not have an observable consequence on the physics of our system. What does this mean? This means that the exchange of two indistinguishable particles must be a symmetry of our system! In nature we find that this manifest itself in two ways: either the wave function of two indistinguishable particles is multiplied by minus one when we exchange the particles, or the wave function is unchanged when we exchange the particles. Indistinguishable particles which obey the first rule are called fermions and those that obey the later rule are called bosons. All of the fundamental particles we know today are either fermions or bosons.
Often, in systems made up of many interacting systems, we find that excitations in these systems have many of the properties of normal particles. Such entities we call quasiparticles. Quasiparticles themselves will be (effectively) indistinguisable, and so too will posses a symmetry under exchange of their positions. An interesting question to ask is what happens to quasiparticles under exchange of their positions. Well, what we find experimentally is that the quasiparticles are almost always fermions or bosons. And of course, what is interesting is the fact that I just said “almost always.” There are cases where we think that the quasiparticles obey rules different from fermions and bosons.
Now a little detour to explain what these different quasiparticles are. Suppose you swap two particles by moving each half way around a circle. Suppose you are viewing this from a fixed point so that you see that the particles are swaped in a clockwise direction. Compare this operation to the process of swapping the particles by moving them half way around the circle in the counterclockwise direction. Are these different processes (that is can we distinguish between these two processes?) Draw an line between the starting points of the two particles. Now rotate the circle about this axis. We now see that we can continuosly deform the clockwise process into the counterclockwise process. We say that in three spatial dimensions, these two processes are topologically indistinguishable. But now imagine that we live in a world with two spatial dimensions. In such a world we can’t perform the above trick. We can’t rotate about that axis between the two particles because that would take us out of our two (spatial) dimensional world. What does this mean? Well in three spatial dimensions, we see that the symmetry which concerns us for indistinguishable particles is that of the symmetric group, where we just permute labels. But in two dimensions, swapping in the counterclockwise direction is different than swapping in the clockwise direction. This means that the symmetry of swapping particles is no longer the symmetric group, but instead is a group called the braid group. If we track the worldlines of particles in two spatial dimensions plus one time dimension, these paths will “braid” each other.
Now back to the story at hand. I said that particles are indistinguishable and so when we swap them this should have no observable consequence. And I said that all fundamental particles and most quasiparticles did this by multiplying the wave function by plus or minus 1. But you might ask why isn’t it possible that we can multiply the wave function by some other phase: say [tex]$e^{i theta}$[/tex]? Well, we know that swapping clockwise around our circle and counterclockwise around our circle are inverse operations of each other. Thus if going clockwise around our circle gave us a phase of [tex]$e^{itheta}$[/tex] then going counterclockwise around our circle should give us a phase of [tex]$e^{-itheta}$[/tex]. But in three spatial dimensions we saw that these two processes were topologically equivalent. This means that [tex]$e^{i theta}=e^{-i theta}[/tex]. But this is only true of [tex]$theta$[/tex] is an integer multiple of pi. Indeed we see that in three spatial dimensions the phase can only be plus or minus one when we swap these particles (there is also the possibility of parastatistics where one uses an internal degree of freedom and obtains higher dimensional representations of the symmetric group. But there is a way to make particles which obey parastatistics to look like they are composites made up of fermions or bosons and hence there is a good reason to say that in three spatial dimensions there are only fermions and bosons.) Now what is cool is that as we argued above, the argument we gave above falls appart for particles in two spatial dimensions. The clockwise and the counterclockwise swap are different in two spatial dimensions, so there is no requirement that [tex]$e^{i theta}=e^{-i theta}[/tex].
This leads one to the postulate that in two spatial dimensions, particles can obey statistics where swapping the particles results in multiplying the wave function by a phase factor [tex]$e^{i theta}$[/tex] where [tex]$theta$[/tex] is not an integer multiple of pi. Such particles are called anyons (the name was coined by Frank Wilczek in 1982. For an interesting interview of Wilczek and anyons see here) Of course we don’t live in a world with two spatial dimensions (we are not Flatlanders, eh?) so one might think that the possibility of anyons existing is nill. Oh well, interesting theory, but no practical applications. Right? Nope.
Remember that we also know that in many body systems there are quasiparticles which act very much like normal particles. And in many body systems we can imagine confining these systems such that they are (effectively) two dimensional. One way to do this is to use semiconductor technology to create very thin flat interfaces of differing materials. Then, by applying an electric field perpendicular to these layers, you create a potential well which can confine electrons along this perpendicular direction. If you cool the system down, then only the lowest energy level of this perpendicular direction will be occupied and the electron(s) will behave as two dimensional objects. So in such systems we might hope that there are quasiparticles which exhibit anyonic properties. Do such systems exist? To answer this question we need another detour. This time we need a detour into the quantum Hall effect.
The Hall effect is a classical effect which you learn about when you first learn classical electrodynamics. Suppose you apply a voltage across two edges of plate. Current will move between these two edges. Now if we apply a magnetic field perpendicular to this plate, the electrons moving along this one direction will experience a Lorentz force in the plate perpendicular to the current. This results in a voltage drop across the plate perpendicular to the current direction. This is the Hall effect. Equilbrium for this setup will occur when the charge buildup on the edges parallel to the current produces an electric field which exactly balances the applied magnetic field. A simple calculation shows that the Hall conductance for this setup is [tex]{ne over B}[/tex] where [tex]$n$[/tex] is the number density for the current carriers, [tex]$e$[/tex] is the electric charge, and [tex]$B$[/tex] is the magnetic field strength. The resistivity, which is the inverse of the conductance, in the Hall effect varies linearly with the applied magnetic field.
In 1980 von Klitzing discovered that if one took one of the two dimensional quantum systems described above, applied a strong magnetic field (a few Tesla) and cooled the system to a few Kelvin, then the Hall resistence no longer varies linearly with the applied magnetic field. In fact the resistence showed a series of steps as a function of the strength of the applied magnetic field. This effect is known as the integer quantum Hall effect. How to explain this effect? Well when we apply a perpendicular magnetic field to our two dimensional system, this results in a change in the energy levels of the system. In particular what happens is that instead of having a continuous set of allowed energy levels for the electrong gas, the levels are now quantized into different discrete (highly degenerate) energy levels. These energy levels are seperated by an amount of energy given by Planck’s constant times the cyclotron frequency of the electrons (the cyclotron frequence is [tex]$omega_C={eB over m}$[/tex], where [tex]$m$[/tex] is the (reduced) mass of the charge carrier. This is the frequency which an electron will circle at in an applied magnetic field.). Now recall than in an electron gas at zero temperature, we fill up the energy levels up to the Fermi energy. But now apply a perpendicular magnetic field, and the different energy levels (called Landau levels) will fill up. But there will be cases where the Fermi energy lies in a gap between the Landau levels. Thus in order for electrons to scatter out of the filled energy levels, they must overcome this energy gap. But at low temperatures they can’t do this and so there is no scattering. Varying the magnetic field moves the spacing between the energy levels. Thus over a range of values where the Fermi energy is in between the Landau levels the Hall resistance will not change. This is the origin of those plataues observed by von Klitzing. We can define a quantity, called the filling factor which tells us how full each Landau level is. At integer values of the filling factor we will observe the effects of the gapped Landau level. This is why it is called the integer quantum Hall effect.
Continuing with our story, in 1982, Stormer and Tsui performed a cleaner version of von Klitzing’s experiment and observed that not only is their a quantum Hall effect for integer values of the filling factor, but that there is also a quantum Hall effect for fractional values of the filling factor. Amazingly these were at very simple integer fractions like 1/3, 1/5, 2/5, etc. For the integer quantum Hall effect, we are essentially dealing with a theory with quasiparticles which are weakly interacting. But for the fraction quantum Hall effect, it was soon realized that the effect must arise from an effect of strongly interacting particles. In 1983 Robert Laughlin put forth a theory to explain the fractional quantum Hall effect by introducing an famous anstatz for the wavefunction of this system which could succesfully explain the plateaus in the fractional quantum Hall effect (well, further modifications were needed for higher filling factor effects.) Now what is interesting, and getting back to our main story, is that this wavefunction has quasiparticle excitations which are anyons! In fact, these excitations would not only behave like they had anyonic statistics, but would also behave like they had fractional values of their charge.
Now the question arises, well the theory explaining the fractional quantum Hall effect has anyonic quasiparticles, but has the effect of these fractional statistics ever been observed. Well there were early experiments which were consistent with such an interpretation, but a really convincing experiment which would directly verify the fractional statistics has never been performed. That is until recently. In Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics by F. E. Camino, Wei Zhou, and V. J. Goldman (all from Stony Brook University) the authors desribe an experiment in which a very cool interferometer experiment is used to directly verify the fractional statistics of the quasiparticle excitations in the fractional quantum Hall effect! This is very cool: the direct observation of fractional statistics!
To me, this whole story, from the theoretical ideas of differing statistics in two dimensions, to the physics of many body strongly interacting systems, and finally to the design of clever, hard experiments to verify the theory behind this strange physics, is one of the most beautiful results in modern physics. Interesting it is also possible that there are anyons which obey nonabelian statistics. This means (roughly) that the wavefunction is not multiplied by a phase under exchange (which is a process which commutes for all such exchanges), but instead another degree of freedom is multiplied by a matrix (so that the noncommutative nature of the braid group is directly realized.) There are some theories of the fractional quantum Hall effect which suggest that these particles might be nonabelian anyons. A complete discussion of nonabelian anyons would lead us on another fascinating story (see John Preskill’s notes on topological quantum computing for an awesome introduction to this subject.) But what is even cooler to ponder is that the experiment preformed in the above article is bringing us one step closer to the possibility that even these strange particles with even stranger statistics may be tested in a similar beautiful experiment. Now that would be an awesome experiment!