I’m here at the Station Q summer meeting at KITP, not too far from home; my office is across the road.
There’s an amazing spectrum of people here. The first of this morning’s talks was by Loren Pfeiffer from Bell Labs, talking about the difficulties of fabricating extremely high mobility GaAs samples — this is the magic stuff in which the fractional quantum hall effect is seen. The second was Kirill Shtengel, explaining unitary braided tensor categories to everyone, so that he could describe how one might perform interferometry experiments on these same quantum hall samples!
I probably can’t say much useful about Loren’s talk — he showed lots of graphs from experiments, and worried about whether nitrogen impurities in the channel were coming from the boron nitride spacers around tantalum wires and so on. On the other hand, if it weren’t for this guy and his friends, at the fifth root of unity would just be some quantum group, rather than a real gadget, sitting on a lab bench somewhere in New Jersey…
Okay, so I can’t quite back that up. The idea is that the physics of quantum hall systems are described by Chern-Simons theory; for each “plateau”, there should be some quantum group, and some root of unity, that tell you everything you need to know. The big question for now is “Which ones?”.
The different regimes or plateaus of the fractional quantum hall effect are named by their “filling fraction”. There’s a bunch of lower “abelian” plateaus, such as , the Laughlin state, for example, with its associated Nobel prizes. Then there are some less well understood ones, 5/2 and 12/5 and 7/4 and so on, all still difficult to play with in the lab. The 5/2 state isn’t too bad, and the consensus is that the right quantum group here is “su(2) at level 2”. Here “level 2” means “at the first 4th root of unity”. (Actually, take that with a grain of salt — first of all there’s some variation in conventions, so perhaps I mean an 8th root. In fact, if you think the dimension of the standard representation of su(2) is , it’s definitely the 8th root. Also, I might just have the translation of levels, which are really about conformal field theory, to roots of unity wrong. I’m using , with k the dual Coxeter number.) You get to identify the quasiparticles in the system with the surviving irreducible representations. The physicists call them 1, and , and they satisfy the rules you’d expect from your favourite tensor product rule in root of unity land: , and .
While this is interesting, it’s not really interesting: while the braiding is nonabelian, it’s not dense. What does this mean? Well, taking the Hilbert space for n particles (be careful — it’s not just the usual tensor power of the space for a single particle!) you can ask if the image of is dense in the unitary group of that Hilbert space. We really want braidings that are dense in this sense, so that we can try to do ‘topological quantum computing’ — which in this guise means approximating our unitary quantum algorithms by words in the braid group, and running algorithms by honestly braiding the right sorts of particles around each other in the lab. You’ll hear people say that a FQH plateau is “universal for quantum computing” in this case. However at 5/2, the image of the braid group is finite. (There is a possible fix, however.)
The next plateau, , is, conjecturally, really interesting. Everyone expects that it’s “so(3) at level 3”, which means roughly the same thing as “the subcategory of the representations of su(2) at a fifth root of unity generated by the 3 dimensional representation”. Again, I might have that “fifth root” part wrong, (possibly even in all conceivable conventions), but it’s easy to work out which one I really want — the underlying fusion algebra of this category is just , with that being the 3 dimensional representation of su(2). And if this really is the right description, then this level is universal. It turns out you can also get the same thing, as a unitary braided tensor category, by looking at at whichever root of unity means that only the trivial and the (formerly) 7-dimensional representation survive. (There’s yet a third way too, but I can’t remember.)
Anyway, Kirill’s talk was about the (theory of the) interferometry experiments you might perform to see that quantum group descriptions of the quantum hall states really are the right ones. Or at least that these states really are nonabelian!
And his conclusion: “None, at this point. Waiting for the experiments.”