What are anyons?

Because of the time lag associated with scientific publishing, there are a number of cool condensed matter results coming out now in the midst of the coronavirus impact.  One in particular prompted me to try writing up something brief about anyons aimed at non-experts.  The wikipedia article is pretty good, but what the heck.

One of the subtlest concepts in physics is the idea of "indistinguishable particles".  The basic idea seems simple.  Two electrons, for example, are supposed to be indistinguishable.  There is no measurement you could do on two electrons that would find different properties (say size or charge or response to magnetic fields).  For example, I should be able to pop an electron out of a hydrogen atom and replace it with any other electron, and literally no measurement you could do would be able to tell the difference between the hydrogen atoms before and after such a swap.  The consequences of true indistinguishability are far reaching even in classic physics.  In statistical mechanics, whether or not a collection of particles and that same collection with two particles swapped are really the same microscopic state is a big deal, with testable consequences.

In quantum mechanics, the situation is richer.  Let's imagine that the only parameter that matters is position.  (We are going to use position as shorthand to represent all of the quantum numbers associated with some particle.)  We can describe a two-particle system by some "state vector" (or wavefunction if you prefer) \( | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle\), where the first vector is the position of particle 1 and the second is the position of particle 2.  Now imagine swapping the two particles.   After the swap, the state should be \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle\).  The question is, how does that second state relate to the first state?  If the particles are truly indistinguishable, you'd think \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle =  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \). 

It turns out that that's not the only allowed situation.  One thing that must be true is that swapping the particles can't change the total normalization of the state (how much total stuff there is).  That restriction is written  \( \langle \psi (\mathbf{r_{2}},\mathbf{r_{1}}) | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle = \langle \psi (\mathbf{r_{1}},\mathbf{r_{2}}) | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \).  If that's the most general restriction, then we can have other possibilities than the states before and after being identical.

For bosons, particles obeying Bose-Einstein statistics, the simple, intuitive situation does hold.  \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle =  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \).

For fermions, particles obeying Fermi-Dirac statistics, instead  \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle = -  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \).  This also preserves normalization, but has truly world-altering consequences.  This can only be satisfied for two particles at the same position if the state is identically zero.  This is what leads to the Pauli Principle and basically the existence of atoms and matter as we know them.

In principle, you could have something more general than that.  For so-called "abelian anyons", you could have the situation  \( | \psi (\mathbf{r_{2}},\mathbf{r_{1}}) \rangle = (e^{i \alpha})  | \psi (\mathbf{r_{1}},\mathbf{r_{2}}) \rangle \), where \(\alpha\) is some phase angle.  Then bosons are the special case where \(\alpha = 0\) or some integer multiple of \(2 \pi\), and fermions are the special case where \(\alpha = \pi\) or some odd multiple of \(\pi\).   

You might wonder, how would you ever pick up weird phase angles when particles are swapped in position?  This situation can arise for charged particles restricted to two dimensions in the presence of a magnetic field. The reason is rather technical, but it comes down to the fact that the vector potential \(\mathbf{A}\) leads to complex phase factors like the one above for charged particles.  

This brings me to this paper.  Anyons have been deeply involved in describing the physics of the fractional quantum Hall effect for a long time  (see here for example).  It's tricky to get direct experimental evidence for the unusual phase factor, though.  The authors of this new paper have been basically doing a form of particle swapping via a scattering experiment, and looking at correlations in where the particles end up (via the noise, fluctuations in the relative currents).  They do indeed see what looks like nice evidence for the expected anyonic properties of a particular quantum Hall state.  

(There are also "nonabelian" anyons, but that is for another time.)