Unified Fock space representation of fractional quantum Hall states

Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin, Moore-Read, and Read-Rezayi wave functions, belong to a special class of orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant). This fundamental observation allows one to point out two different recurrence relations for the coefficients of the permanent (Slater) decomposition of the bosonic (fermionic) states. Here we provide an explicit Fock space representation for these wave functions by introducing a two-body squeezing operator which represents them as a Jastrow operator applied to reference states, which are, in general, simple periodic one-dimensional patterns. Remarkably, this operator representation is the same for bosons and fermions, and the different nature of the two recurrence relations is an outcome of particle statistics.

Figure 1

(a) Each site $0,1,2,\cdots$ is an angular momentum state in the LLL. A configuration of occupancies corresponds to a partition. Here the lattice configuration on the left corresponds to the partition $(6,1,0)$. (b) Lattice and partition representation of one of the possible squeezings of the partition $(6,1,0)$. In Fock space, this corresponds to the action of the two-body operator $a_2^{†}{a}_5^{†}{a}_6{a}_1$. (c) The squeezing $(6,1,0)\to (4,2,1)$ shows the power of the Fock space representation. Indeed using the operator $a_2^{†}{a}_4^{†}{a}_6{a}_0$, minus signs arising in the fermionic case are automatically taken into account, making the unification of the bosonic and fermionic FQH states made in Eq. (16) possible.

Figure 2

The repeated action of the squeezing operator of Eq. (17) on the bosonic and fermionic Laughlin states with (a) $\nu =1/2$ and (b) $\nu =1/3$, with three particles. Each state is specified by the occupancy numbers and each arrow represents the action of a squeeze. The operator in Eq. (17) automatically takes into account the possibility of multiple occupancies of the bosonic case.

Figure 3

The different factors ${\mu }_i-{\mu }_j$ (bosons) and ${\theta }_i-{\theta }_j$ (fermions) in Eq. (29) can be understood graphically: squeezings (a) and (b) produce the same physical state because the two particles which are colored in green and red are actually indistinguishable. However, due to the fact that in (b) there is a crossing between particles, the two states differ for a phase which depends on the statistics: $+1$ for bosons, $-1$ for fermions. The squeezing operator defined in Eq. (17) automatically accounts for the difference between bosons and fermions, as explicitly shown in Appendix pp2.