203. Quantum Hall effect. One open question

I cannot finish discussing quantum Hall effect without explaining why, after all those years of study, Laughlin wavefunction and composite fermions, it still remains interesting for a condensed matter theorist.

The reason is existence of the state with , the only state with even denominator of the filling factor

Let me put the story again into historical context. First, Laughlin introduced his famous wavefunction that nicely explained FQHE states with filling factors , where is odd. As far as I understand, the logic was the following: noting that FQHE states seem to correspond to fractionally charged quasiparticles, one introduces a probe wavefunction based on symmetry arguments (the ones that Lubos used in his comment) and starts to apply the variational principle to it looking for the state with lowest energy. One inevitably finds the Laughlin wavefunction (corresponding to the energy lower than the energy of the trivial state). Second, since it was still necessary to explain FQHE states with filling factors other than , it became clearly necessary to generalize the Laughlin wavefunction. Haldane and Galperin did exactly this, but the hierarchy following from their ansatz turned out to contain also states with filling factors that are not observed in experiment. Later, the concept of composite fermions came into play – and it explained almost all the FQHE states, all, except one – the state with filling factor , the only state with even denominator of that exist in a single layer 2-dimensional electron gas.

The puzzle of this state remains unsolved. Currently, there is belief that is a fully spin polarized FQHE state described by so called Moore-Read wavefunction. Some people also think that it might describe anyons with non-abelian fractional statistics. Even if so, it remains absolutely unclear why all other FQHE states correspond to abelian statistics as well as why there are no other states with even denominator of the filling factor. I hope that somebody among you, who read this now, will crack the puzzle of the state, bring the story about fractional quantum Hall effect to the whole new level and, who knows, may be get her Nobel Prize if politics permits.