Beyond paired quantum Hall states: parafermions and incompressible   states in the first excited Landau level
  The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k+1-body interactions, for all integers k > 0. The remarkably simple wavefunctions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2 case is the Pfaffian. For k > 1, the quasiparticle excitations of these systems are expected to possess nonabelian statistics, like those of the Pfaffian. For k=3, these ground states have large overlaps with the ground states of the (2-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors \nu=2+3/5, 2+2/5. 
N. Read and E. Rezayi