Structure of spinful quantum Hall states: A squeezing perspective

We provide a set of rules to define several spinful quantum Hall model states. The method extends the one that is known for spin-polarized states. It is achieved by specifying an undressed root partition, a squeezing procedure, and rules to dress the configurations with spin. It applies to both the excitationless and the quasihole states. In particular, we show that the naive generalization where one preserves the spin information during the squeezing sequence may fail. We give numerous examples such as the Halperin states, the non-Abelian spin-singlet states, or the spin-charge separated states. The squeezing procedure for the series $(k=2,r)$ of spinless quantum Hall states, which vanish as $r$ powers when $k+1$ particles coincide, is generalized to the spinful case. As an application of our method, we show that the counting observed in the particle entanglement spectrum of several spinful states matches the one obtained through the root partitions and our rules. This counting also matches the counting of quasihole states of the corresponding model Hamiltonians, when the latter are available.

Figure 1

The configurations enumerating the number of $S=0$ states in the tensor product of six spin-$1/2$ particles.

Figure 2

The configurations enumerating the number of $S=1$ states in the tensor product of six spin-$1/2$ particles.

Figure 3

The particle entanglement spectrum for the bosonic Haffnian state with $N=10$ particles, keeping $N_A=5$ particles. The $L_{z,A}$ degeneracy is due to the multiplet structure associated with $L_A^2$. The counting per value $L_{z,A}$ sector exactly matches the corresponding number of quasihole states for 5 particles and 10 added flux quanta.

Figure 4

The particle entanglement spectrum for the bosonic Haffnian state with $N=10$ particles, keeping $N_A=5$ particles. Only the entanglement levels of the highest weights are shown.

Figure 5

The particle ES for the $N=12$ bosonic AS state, with $N_A=6$.

Figure 6

The particle entanglement spectrum for the bosonic spin-permanent state SBper, with $N=6$ particles, keeping (a) $N_A=2$ and (b) $N_A=3$ particles.

Figure 7

The particle entanglement spectrum for the bosonic spin-permanent state SBper, with $N=8$ particles, keeping $N_A=4$.