Quantum Hall wave functions based on $S_3$ conformal field theories

We construct a family of quantum Hall Hamiltonians whose ground states, at least for small system sizes, give correlators of the $S_3$ conformal field theories. The ground states are considered as trial wave functions for quantum Hall effect of bosons at filling fraction $\nu =3/4$ interacting either via delta function interaction or delta function plus dipole interaction. While the $S_3$ theories can be either unitary or nonunitary, we find high overlaps with exact diagonalizations only for the nonunitary case, suggesting that these wave functions may correspond to critical points, possibly analogous to the previously studied Gaffnian wave function. These wave functions give an explicit example which cannot be fully characterized by their thin-torus limit or by their pattern of zeros.

Figure 1

(Color online) Squared overlap of $N=12$ bosons at $\nu =3/4$ exact wave function with the $S_3$ trial wave function as a function of the tuning parameter $\theta$. Solid curve shows the overlap of the trial wave functions with the exact ground state of bosons interacting via repulsive delta function interaction. Dashed curve is the overlap with the exact ground state of bosons interacting via repulsive delta function plus dipole interaction where the amount of dipole interaction is set such that $V_2/V_0=0.2$. Inset: overlap at $\theta =0$ as a function of amount of dipole interaction $V_2/V_0$. The vertical axis of the inset is aligned with the vertical axis of the main plot.