Topological Entanglement and Clustering of Jain Hierarchy States

We obtain several clustering properties of the Jain states at filling (k/2k+1): they are a product of a Vandermonde determinant and a bosonic polynomial at filling (k/k+1) which vanishes when k+1 particles cluster together. We show that all Jain states satisfy a 'squeezing rule' which severely reduces the dimension of the Hilbert space necessary to generate them. We compute the topological entanglement spectrum of the Jain ν=(2/5) state and compare it to both the Coulomb ground state and the nonunitary Gaffnian state. All three states have a very similar 'low-energy' structure. However, the Jain state entanglement 'edge' state counting matches both the Coulomb counting as well as two decoupled U(1) free bosons, whereas the Gaffnian edge counting misses some of the edge states of the Coulomb spectrum.