Once you have an organized ground state, it is a natural question to ask what the low energy “defects” of this ground state, (the quasiparticles) look like. Indeed, in most cases, it is these quasiparticles that determine the interesting physical properties of the phase of matter in the first place. All of the particles have fallen into line perfectly making a featureless background, and what you notice most in the experiments are the few regions where something different is going on. A next question to ask is what happens when you have a lot of these defects. Can the defects now start forming their own organized collective – their own Borg?

At the Quantum Hall workshop at NORDITA this month there has been a lot of discussion of what kind of condensates, or new phases of matter, can form from collections of quasiparticles in fractional quantum Hall states. This is an old question that dates back to the very earliest days of quantum Hall effect. As many people reading this might already know, very shortly after the discovery of the nu=1/3 fractional quantum Hall effect, Bob Laughlin gave a beautiful theoretical explanation of how electrons in high magnetic field can condense into a new quantum phase of matter (a Borg of electrons), thus explaining the experiment. However, very soon thereafter, additional quantum Hall effects were discovered (the 2/3 effect, the 2/5 effect, and so on). Laughlin’s theory did not fully explain these. It was Halperin and Haldane who realized that the defects (the quasiparticles) of the nu=1/3 effect can themselves organize, forming further new phases of matter. The resulting picture was a recursive construction of defects condensing then new defects forming within these new condensates.

So why revisit this issue now? Well, the new twist is an entirely new class of more complex and interesting quantum Hall states – the so-called “nonabelian” phases or “nontrivial topological” phases (drawing a distinction that all of the abelian phases are now considered “trivial”). In these cases, the quasiparticles, in addition to carrying charge (and fractional statistics), also carry interesting topological quantum numbers. It is not so obvious how such a thing can form a condensate at all, or whether it would want to do so.

There have been several approaches to addressing this problem. The first set of approaches attempt to condense the nonabelian anyon by forming a topologically trivial combination of quasiparticles and then condensing the combination in the same spirit as the old Halperin-Haldane hierarchy.

(1) An approach by Bonderson and Slingerland combines a pair of quasiparticles on top of each other in a topologically trivial combination then condenses these pairs. A more recent paper by same authors plus Moller and Feiguin shows some nice numerical data showing that these trial states are actually quite competitive for experimental systems – although from the data I saw, it was not completely convincing that there was any regime in which they clearly were better than more conventional trial states. Nonetheless, they seem to be now in the running as something that needs to be seriously considered.

(2) An approach by Levin and Halperin (neither of them happen to be at this conference) is to form a topologically trivial quantum superposition of states before condensing. (Not surprisingly, the resulting states lose all of their topologically interesting properties after the condensation ). There does not appear to be much experimental or numerical evidence of these states being realized, even for model systems.

(3) A third approach by Hermanns is a bit more confusing to describe. At first I thought that it was probably incorrect, but now I think the construction makes a fair amount sense although there are some pieces of the argument that still seem a bit mysterious to me. I’ve agreed to be on Maria Hermanns’ thesis committee, to be her “opponent” in the Swedish system, which I gather means it is my job to find holes in her arguments, so I’ll be studying this a lot more in the next few months.

(4)In the work of Schoutens and Grosberg a condensate naively looks a bit different. In this case, a condensate is made by forming a maximum density droplet of a particular quasiparticle with nontrivial topological quantum number . This case can be analyzed in great detail – determining not only the details of the condensate (which is a known phase) but also the behavior of the edge separating the mother and daughter states. (See below however, the work of (6) seems to be able to phrase this condensation again as a boson condensing).

And there are yet more approaches. In the above approaches, all of the quasiparticles form liquids. There is another possibility which is that all of the quasiparticles form a solid. Solidification would usually be considered uninteresting from a topological perspective, but here since the quasiparticles carry topological quantum numbers something more interesting can happen.

(5) In this picture discussed by

Finally, there is the world of more abstract nonsense:

(6) And a more abstract discussion of condensation was given by Slingerland and Bais. While perhaps a bit daunting at first, this paper is well worth the effort to read. These authors have constructed a generalized paradigm to describe condensation of one topological phase within another topological phase. The general rule is simply that you have to find a particle that is topologically a boson, then you can condense it. Anything that is not “local” with respect to the boson cannot live within the new phase, and you have to identify any two particles that differ from each other by the bosons. (There is a subtlety having to do with particle branching that I will not explain here). Pretty much all of the above cases can be described within this formalism in one way or another. Further, coset TQFTs can be described nicely within this formalism too (which I find very pretty). The down side, as in any abstract nonsense, is the generality is frequently a disadvantage as much as an advantage. Since you can describe pretty much anything, it does not give you hints as to what thing to expect.

At any rate, there are a whole bunch of ways to describe condensation of topological phases within topological phases. Seems like a popular thing to be studying right now. Resistance is futile….

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