The thermodynamic limit and the FQHE

I'm taking Tony Leggett's course on two-dimensional physics; it's the first time I've thought hard about the fractional quantum Hall effect, the quantum Hall and spin-Hall effects, fractional statistics, and related "topological" phenomena. I think I'm finally beginning to understand what's behind these effects, and it's essentially a rather subtle business with adiabaticity. Hopefully I'll straighten all of this out in due course; for now, here are two papers on the ground-state degeneracy of the FQHE:

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http://prola.aps.org/abstract/PRB/v41/i13/p9377_1
Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces
X.-G. Wen and Q. Niu
Phys. Rev. B 41, 9377 - 9396 (1990)

The fractional quantum Hall (FQH) states are shown to have q̃^g-fold ground-state degeneracy on a Riemann surface of genus g, where q̃ is the ground-state degeneracy in a torus topology. The ground-state degeneracies are directly related to the statistics of the quasiparticles given by θ=p̃π/q̃. The ground-state degeneracy is shown to be invariant against weak but otherwise arbitrary perturbations. Therefore the ground-state degeneracy provides a new quantum number, in addition to the Hall conductance, characterizing different phases of the FQH systems. The phases with different ground-state degeneracies are considered to have different topological orders. For a finite system of size L, the ground-state degeneracy is lifted. The energy splitting is shown to be at most of order e-L/ξ. We also show that the Ginzburg-Landau theory of the FQH states (in the low-energy limit) is a dual theory of the U(1) Chern-Simons topological theory.

http://prola.aps.org/abstract/PRL/v66/i6/p806_1
Fractional quantum Hall effect and multiple Aharonov-Bohm periods
D.J. Thouless and Y. Gefen
Phys. Rev. Lett. 66, 806 - 809 (1991)

An arrangement for obtaining Aharonov-Bohm oscillations of basic periodicity qh/e (q>1) is discussed. The relaxation towards h/e periodicity is characterized by a decay time exponential in the system size at zero temperature, and linear in this size at finite temperature.

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The Thouless-Gefen paper is particularly good, see esp. Fig. 1. [I think the following text is unintelligible without the figure but should make sense once you look at it.] The idea is to consider the FQHE in a CD-shaped geometry (called, for obscure reasons, a Corbino disk) with a fixed field B through the "substance" of the disk -- which creates the FQHE -- and a variable flux F (sorry about the notation, I really ought to install the tex widget...) threading the hole. Quantum mechanics says that the ground state energy should be periodic in the flux threading the hole, with a period of one flux quantum; Cooper pairs give you half this period because they have charge 2e, whereas a system with "fractionally charged" excitations (say q = e/3, the classic Laughlin state) gives you three times this period, which appears to contradict a basic quantum mechanical result called the Byers-Yang theorem. Ultimately this turns out to be about level crossings (Fig. 1 of Thouless-Gefen): there are three ground states that have vanishingly small matrix elements mixing them; if one increased the flux F vanishingly slowly, one would track the ground state of the system, and have 1/e-periodic behavior; if one increased the flux rapidly, one would go straight up the original curve, not notice the level crossings, and get no periodicity at all ("physically" this corresponds to continually adding quasiparticles); in practice, one goes "rapidly" as far as the second level crossing and "adiabatically" with respect to that (b'se there's a much bigger avoided-level-crossing there) so one comes back to the ground state with 3/e periodicity.

A key corollary, I think, is that fractional charge (and therefore fractional statistics) are a consequence of non-avoided and avoided level crossings when you "slowly" move a particle around another -- i.e. a consequence of the order in which you take the limit of infinite system size and the limit of moving a particle around another (or ramping up a flux) infinitely slowly. This does not mean they aren't "real," of course, since phase transitions, which are evidently real, are also ultimately a consequence of the order in which one takes limits. It appears to be true that in all known cases of fractional statistics, at least of the abelian kind (another example is Kitaev's toric code model), one has this sort of ground-state degeneracy.