thanks for the explanations! I guess the ultimate question is why FQHE with odd and even q belong to different universality classes (why they are so amazingly different in other words), but probably nobody has an answer to this question nowadays…

Cheers,
Dmitry.

Let me answer you by remembering the general idea behind this line of research..The point in this research is to characterize distinct phases of matter. I recall that, at a given filling factor, you can find FQH states described in the asymptotic low-energy limit, and thus independent of the microscopic details details of the effective Hamiltonian and of the wave-function, by different universality classes,i.e. ground-state quantum number, properties of excitations etc etc.
For instance it turns out that the filling 2+1/3 (so the first LL at filling 1/3)the Laughlin states have not a good overlap with the exact ground state. It has been then necessary to investigate other trial wave-functions, as the paired ones which form Lauglins States by pairing electrons, which are representative of other universality classes.

All this just to say that, as in the case of the study of the critical point of statistical models, I cannot in general say, on the basis of a the microscopic hamiltonian, which is the universality class describing my system. So one tries to classify the possible classes on the basis of general “symmetries”, as the properties to be zero-energy eigenstate of specific pseudi-potential hamiltonian..

thanks for the great understandable post! Let me ask you a couple of naive questions:

1) if we focus on filling factor , which state is more energetically favorable – Laughlin or Moore-Read? (I thought that Laughlin’s is proved to be the lowest energy one by variational methods)

2) why should we ultimately focus only on three-body projection Hamiltonian? Is the logic behind 3-body that
it seems to be possible to describe non-abelian statistics within 3-body Hamiltonian states,therefore, it might be relevant for describing FQHE with where is even?

3) related to (1) and (2): if Laughlin w.f. is energietically favorable for odd , while Moore-Read – for even , why 3-body interactions are important for even ?

Cheers,
Dmitry.