Construction of a paired wave function for spinless electrons at filling fraction ν=2∕5

We construct a wave function, generalizing the well-known Moore-Read Pfaffian, that describes spinless electrons at filling fraction ν=2∕5 (or bosons at filling fraction ν=2∕3) as the ground state of a very simple three body potential. We find, analogous to the Pfaffian, that when quasiholes are added there is a ground state degeneracy which can be identified as zero modes of the quasiholes. The zero modes are identified as having semionic statistics. We write this wave function as a correlator of the Virasoro minimal model conformal field theory M(5,3). Since this model is nonunitary, we conclude that this wave function is likely a quantum critical state. Nonetheless, we find that the overlaps of this wave function with exact diagonalizations in the lowest and first excited Landau level are very high, suggesting that this wave function may have experimental relevance for some transition that may occur in that regime.