Tuesday, February 7, 2012

Ready to start

"So the rule is that composite objects, in circumstances in which the composite object can be considered a single object, behave like Fermi particles or Bose particles, depending on whether they contain an odd number or an even number of Fermi particles." - Feynman's Lectures (Vol. III, ch. 4)

So another blog. Another science blog, at that. Don't we have enough of those? Perhaps yes to the former, but I'd argue it's impossible to have enough of the latter. But another physics blog? Well, there's a glut of high energy, particle, and astrophysics blogs, since those are the sexy topics that luminaries like Stephen Hawking and Morgan Freeman make television shows about. But condensed matter physics blogs are harder to come by, so I don't see any harm in adding another to the pile. In particular, I think getting perspectives from as many demographics as possible is a good thing, so that's my excuse for creating a blog while still a lowly graduate student who should be reading right now.

"Anyons?" you may ask, "what're those?" You may already know about bosons and fermions, the two varieties of fundamental particles (advice for non-experts: any word you don't recognize that ends in "-on" or "-ons" probably refers to some type of particle). Fermions are electrons, quarks, and the various other building blocks of matter. Bosons are the particles that carry the three-plus-one fundamental interactions, and the Higgs if it exists. What's the difference? Why the two classifications?

In quantum mechanics, we must be careful to not pretend we know more than we actually do. With regards to particles, if we can only specify them by their quantum numbers (spin, charge, energy etc.), then we cannot assume any additional properties of them. This means that given two electrons, for instance, we must assume they are identical in all aspects other than ones we know are different. And if two particles are identical, that means we can switch their locations and all physical predictions should be the same - which means the probabilities of all measurement outcomes are unchanged. In elementary wave mechanics, the probability of detecting a particle at a location is given by the amplitude-squared of the wavefunction evaluated at that location. This notion can be generalized to multiple particles: instead of a wavefunction \psi(x), we have \psi(x_1, x_2) that is related to the probability of detecting a particle at x_1 AND a particle at x_2. So if both particles are identical, then |\psi(x_1,x_2)|^2 better be the same as |\psi(x_2,x_1)|^2. What this means is that the wavefunction itself must change by some complex phase e^{i \delta} when we interchange x_1 and x_2 (since |e^{i \delta}|^2 = 1). However, we additionally require that two interchanges get us back to where we started (since this operation is equivalent to doing nothing), hence e^{2i \delta} = 1, meaning \delta must be either 0 or \pi (adding multiples of 2\pi to these values obviously work as well, but those result in identical complex phases and don't tell us anything new).

That is, exchanging identical particles can either leave the wavefunction unchanged (\delta = 0) or multiply it by negative one (\delta = \pi). The former are bosons, the latter are fermions. The statistical consequences of these facts lead to multitudes of interesting phenomena, but from a theoretical perspective, fermions and bosons are boring. They're old hat; we understand them pretty well (especially since the spin-statistics theorem of relativistic quantum field theory).

It is an experimental fact that all known fundamental particles are either fermions or bosons, and as the quote from Feynman at the beginning indicates, it was pretty much accepted that any combinations of these fermions and bosons would have to fall under one umbrella or the other as well. So why gripe about their boring-ness if they're all we have? Shouldn't we be content?

Well, it turns out fermions and bosons aren't the whole story. Sometime around the late-seventies/early-eighties theorists began to examine what it meant to really "exchange" particles, and they discovered something remarkable: in two dimensions, a whole cornucopia of different particle types are possible, generally known as anyons since that phase \delta from above can take on literally (said like Chris Traeger) any value. We'll talk more about this interesting subject later. Not only can that complex phase have any value, but more generally the wavefunction can be transformed by a unitary matrix (hence the "non-abelian" variety of anyons, since the matrices do not in general commute).

Of course, it may be objected that we don't live in two dimensions, we live in three! So who cares about these theoreticians' fantasies? This is a valid point, and as I said before before, experimentally all known fundamental particles do what we expect for three dimensions. However, we are able to constrain the particles of our three dimensional world to live in two dimensional surfaces. This is achieved in so-called two-dimensional electron gases or 2DEGs, where electrons become trapped at the interface between two different semiconductor layers, or in a very narrow quantum well, such that there is no transport (fancy word for motion) in the transverse direction (i.e. perpendicular to the surface). Examples of these are GaAs/AlGaAs heterostructures, which are widely studied in mesoscopics, nanoscience, and quantum Hall effects.

So the fundamental electrons, despite being trapped to 2D, are still fermions - we can't change that. However, it turns out that Feynman was wrong (for once): there are certain combinations of these electrons that do have anyonic statistics, because these quasiparticles are fundamentally living in 2D.

I think that's all for now, I do have actual work to get done, but I'll continue this story later.