Generalized Clustered Quantum Hall States

The Read-Rezayi (parafermion) quantum Hall states[1] for bosons can be defined as states where the wavefunction does not vanish when $g$ bosons come together to the same point, but does vanish as $z^2$ as a $g+1$st particle approaches that point. These states can equivalently be defined as the unique ground state of a point contact $g+1$ particle interaction Hamiltonian. Interestingly, the series of Read-Rezayi states appears to describe well the groundstates of rotating Bose condensates with point-contact two body interactions at a series of filling fractions [2]. If one attaches a Jastrow factor to such bose wavefunctions, one obtains fermion wavefunctions that may occur in electronic quantum Hall systems including the ($g=2$) Pfaffian [3] and the ($g=3$) $\nu=13/5$ Read-Rezayi state [1]. In this work, we consider generalized cluster wavefunctions defined by the algebraic manner in which a wavefunction vanishes as $g+1$ particles coalesce. We find Hamiltonians that generate these wavefunctions as their exact ground state. Among this series of states is the previously studied Haffnian wavefunction[4] and a host of states not previously discussed. We catalogue and study the new states and discuss whether any of them might occur in actual physical systems. [1] N. Read and E. Rezayi, PRB{\bf 59}, 8084 (1999). [2] N. R. Cooper, N. K. Wilkin, and J. M. F. Gunn, PRL{\bf 87}, 120405 (2001) [3] G. Moore and N. Read, Nuc. Phys. B{\bf 360}, 362 (1991). [4] D. Green, PhD Thesis.