Quasiparticles and excitons for the Pfaffian quantum Hall state

We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the conformal field theory (CFT) that describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the Fractional Quantum Hall plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the three-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and “squeezing.”

Figure 1

Sketch of the composite fermion mapping in the case of two vortices attached [$m=2$ in Eq. (18)].

Figure 2

(a) The bosonic $\nu =1/2$ Laughlin state in the composite fermion picture. (b) An exciton over this state, with one CF in the second $\Lambda$L and one hole in the L$\Lambda$L. (c) A quasielectron excitation of the Laughlin state: as the number of flux quanta is decreased by one unit, one CF has to occupy the second $\Lambda$L. (d) A quasihole excitation of the Laughlin state: as the number of flux quanta is increased by one unit, there is one hole in the L$\Lambda$L.

Figure 3

Top left: spectra for $N=16$ and two quasielectrons ($N_{\phi }=13$) of the Hamiltonian ${\mathcal{H}}_B^{\left(3\right)}$ in the full Hilbert space (dashes) in the full space of trial two-quasielectron states (crosses) and in the space of trial states that vanish when four particles are brought to the same point (dots). Top right: spectra of ${\mathcal{H}}_B^{\left(2\right)}$ in the same spaces. Bottom left: spectra of ${\mathcal{H}}_F^{\left(3\right)}$ for $N=14$ and two quasielectrons ($N_{\phi }=24$) in the analogous spaces for fermions. Bottom right: spectra of the second LL Coulomb Hamiltonian with $\delta V_1=0.035$ in the same spaces.

Figure 4

Top left: spectra for $N=16$ and four quasielectrons ($N_{\phi }=12$) of the Hamiltonian ${\mathcal{H}}_B^{\left(3\right)}$ in the full Hilbert space in the space of trial four-quasielectron states and in the space of trial states that vanish when four particles positions coincide. Top right: spectra of ${\mathcal{H}}_B^{\left(2\right)}$ in the same spaces. Bottom left: spectra of ${\mathcal{H}}_F^{\left(3\right)}$ for $N=14$ and four quasielectrons ($N_{\phi }=23$) in the analogous spaces for fermions. Bottom right: spectra of the second LL Coulomb Hamiltonian with $\delta V_1=0.035$ in the same spaces.

Figure 5

Spectra for bosons at $N=16$ and $N_{\phi }=14$. Top left: spectrum of ${\mathcal{H}}_B^{\left(3\right)}$ in the full Hilbert space (dashes) and in the space spanned by our one exciton trial states (dots). Bottom left: spectra of ${\mathcal{H}}_B^{\left(2\right)}$ in the same spaces. Top right: spectra of ${\mathcal{H}}_B^{\left(3\right)}$ in the full Hilbert space in the space of our two-exciton trial states (crosses) and in the space of two-exciton trial wave functions, which are zero modes of ${\mathcal{H}}_B^{\left(4\right)}$ (dots). Bottom right: spectra of ${\mathcal{H}}_B^{\left(2\right)}$ in the same spaces.

Figure 6

Spectra for fermions at $N=14$ and $N_{\phi }=25$. Top left: spectra of ${\mathcal{H}}_F^{\left(3\right)}$ in the full Hilbert space (dashes) and in the space spanned by our one-exciton trial states (dots). Bottom left: spectra of the second LL Coulomb interaction with $\delta V_1=0.035$ in the same spaces. Top right: spectra of ${\mathcal{H}}_F^{\left(3\right)}$ in the full Hilbert space, in the space of our two-exciton trial states (crosses) and in the space of two-exciton trial wave functions, which are zero modes of ${\mathcal{H}}_F^{\left(4\right)}$ (dots). Bottom right: spectra of the second LL Coulomb interaction with $\delta V_1=0.035$ in the same spaces.