What are parastatistics?

While I could certainly write more about what is going on in the US these days (ahh, trying to dismantle organizations you don't understand), instead I want to briefly highlight a very exciting result from my colleagues, published in Nature last month.  (I almost titled this post "Lies, Damn Lies, and (para)Statistics", but that sounds like I don't like the paper.)

When we teach students about the properties of quantum objects (and about thermodynamics), we often talk about the "statistics" obeyed by indistinguishable particles.  I've written about aspects of this before.  "Statistics" in this sense means, what happens mathematically to the multiparticle quantum state \(|\Psi\rangle\) when two particles are swapped.  If we use the label \(\mathbf{1}\) to mean the set of quantum numbers associated with particle 1, etc., then the question is, how are \(|\Psi(\mathbf{1},\mathbf{2})\rangle\) and \(|\Psi(\mathbf{2},\mathbf{1})\rangle\) related to each other.  We know that probabilities have to be conserved, so \(\langle \Psi(\mathbf{1},\mathbf{2}) | \Psi(\mathbf{1},\mathbf{2})\rangle = \langle \Psi(\mathbf{2},\mathbf{1}) | \Psi(\mathbf{2},\mathbf{1})\rangle\).   

The usual situation is to assume \(|\Psi(\mathbf{2},\mathbf{1})\rangle\ = c |\Psi(\mathbf{1},\mathbf{2})\rangle\), where \(c\) is a complex number of magnitude 1.  If \(c = 1\), which is sort of the "common sense" expectation from classical physics, the particles are bosons, obeying Bose-Einstein stastistics.   If \(c = -1\), the particles are fermions and obey Fermi-Dirac statistics.  In principle, one could have \(c = \exp(i\alpha)\), where \(\alpha\) is some phase angle.  Particles in that general case are called anyons, and I wrote about them here.  Low energy excitations of electrons (fermions) confined in 2D in the presence of a magnetic field can act like anyons, but it seems there can't be anyons in higher dimensions.

 Being imprecise, when particles are "dilute" -- "far" from each other in terms of position and momentum -- we typically don't really need to worry much about what kind of quantum statistics govern the particles.  The distribution function - the average occupancy of a typical single-particle quantum state (labeled by a coordinate \(\mathbf{r}\), a wavevector \(\mathbf{k}\), and a spin \(\mathbf{\sigma}\) as one possibility) - is much less than 1.  When particles are much more dense, though, the quantum statistics matter enormously.  At low temperatures, bosons can all pile into the (single-particle, in the absence of interactions) ground state - that's Bose-Einstein condensation.   In contrast, fermions have to stack up into higher energy states, since FD statistics imply that no two indistinguishable fermions can be in the same state - this is the Pauli Exclusion Principle, and it's basically why solids are solid.  If a gas of particles is at a temperature \(T\) and a chemical potential \(\mu\), then the distribution function and a function of energy \(\epsilon\) for bosons or fermions is given by \(f(\epsilon,\mu,T) = 1/ (\exp((\epsilon-\mu)/k_{\mathrm{B}}T) \pm 1 )\), where the \(+\) sign is the fermion case and the \(-\) sign is the boson case.  

In the paper at hand, the authors take on parastatistics, the question of what happens if, besides spin, there are other "internal degrees of freedom" that are attached to particles described by additional indices that obey different algebras.  As they point out, this is not a new idea, but what they have done here is show that it is possible to have mathematically consistent versions of this that do not trivially reduce to fermions and bosons and can survive in, say, 3 spatial dimensions.  They argue that low energy excitations (quasiparticles) of some quantum spin systems can have these properties.  That's cool but not necessarily surprising - there are quasiparticles in condensed matter systems that are argued to obey a variety of exotic relations originally proposed in the world of high energy theory (Weyl fermions, Majorana fermions, massless Dirac fermions).  They also put forward the possibility that elementary particles could obey these statistics as well.  (Ideas transferring over from condensed matter or AMO physics to high energy is also not a new thing; see the Anderson-Higgs mechanism, and the concept of unparticles, which has connections to condensed matter systems where electronic quasiparticles may not be well defined.)

Fig. 1 from this paper, showing distribution functions for fermions, bosons, 
and more exotic systems studied in the paper.

Interestingly, the authors work out what the distribution function can look like for these exotic particles, as shown here (fig 1 from the paper).  The left panel shows how many particles can be in a single-particle spatial state for fermions (zero or one), bosons (up to \(\infty\)), and funky parastatistics-obeying particles of different types.  The right panel shows the distribution functions for these cases.  I think this is very cool.  When I've taught statistical physics to undergrads, I've told the students that no one has written down a general distribution function for systems like this.  Guess I'll have to revise my statements on this!