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>~1. The remarkably simple wave functions 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>~2, the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For k=3, these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors =2+3/5, 2+2/5.