Topological Entanglement and Clustering of Jain Hierarchy States

We obtain several clustering properties of the Jain states at filling $\frac{k}{2k+1}$: they are a product of a Vandermonde determinant and a bosonic polynomial at filling $\frac{k}{k+1}$ 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 $\nu =\frac{2}{5}$ 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.

Figure 1

Root partition in angular momentum basis for $\nu =\frac{2}{3},\frac{3}{4},\frac{4}{5}\dots \frac{k}{k+1}$ states can be written as the sum of the Vandermonde determinant partition plus the maximum root partition of the determinant operator of $k$ LL projected to the LLL. The root occupation configuration contains $k$ particles in $k+1$ orbitals $[1^{k}0]$ when deep in the bulk. Close to the north and south pole there are deviations from this rule.

Figure 2

Top: Topological entanglement of the pure Coulomb at $\nu =\frac{2}{5}$. The levels below the light blue line at $\xi \approx 8$ are almost identical to the Gaffnian levels, whereas the levels below the green line are almost identical to those of the Jain state, to within 0.003%–3%. Middle: Topological entanglement spectrum of the Gaffnian state. Bottom: Topological entanglement spectrum of the $N=16$, $\nu =2/5$, Jain state. The low-energy structure of the Jain state (in blue at $\xi \approx 8$) is almost identical to that of the Gaffnian (below the blue line).

Figure 3

Red plots [starting at ${\delta }_0\approx 1$ in (a) and ${\delta }_1\approx 0.5$ in (b)]: Entanglement gap of the Coulomb state for $L_z^{(A)}=80$ (a), $L_z^{(A)}=79$ (b) as a function of added hard-core potential. The entanglement gap is discontinuous (a, b) at similar values of $\delta V_1$ for which the overlap of the Gaffnian and Jain with the Coulomb ground state collapses [see inset (a): overlaps of both Gaffnian and Jain states with the Coulomb ground state as a function of added hard-core interaction $\delta V_1$ for $N=16$ particles]. A phase transition occurs close to $\delta V_1\simeq -0.08$.] The red dashed line is the entanglement gap for the Jain state. Green plots [starting at ${\delta }_0\approx 3$ in (a) and ${\delta }_1\approx 1$ in (b)]: Overlap of the reduced density matrix eigenstates for each of the $L_z^{(A)}=80$, 79 between Coulomb and Gaffnian. The green dashed line is the calculation result between Jain and Gaffnian. Inset (b): Similar results in the two $U(1)$ free boson sector.