Topological $d$-wave pairing structures in Jain states

We discuss $d$-wave topological (broken time-reversal symmetry) pairing structures in unpolarized and polarized Jain states. We demonstrate pairing in the Jain spin-singlet state by rewriting it in an explicit pairing form, in which we can recognize $d$-wave weak pairing of underlying quasiparticles—composite fermions. We find and describe the root configuration of the Jain spin-singlet state and its connection with neutral excitations of the Haldane-Rezayi state, and study the transition between these states via exact diagonalization. We find high overlaps with the Jain spin-singlet state upon a departure from the hollow-core model for which the Haldane-Rezayi state is the exact ground state. Due to a proven algebraic identity we are able to extend the analysis of topological $d$-wave pairing structures to polarized Jain states and integer quantum Hall states and discuss its consequences.

Figure 1

Sketches of the excitation spectra of the Jain spin-singlet (left panel) and Haldane-Rezayi (right panel) states with the respective root configurations of the ground states.

Figure 2

Excitation gap and overlaps of the ground state for a two-body interaction Hamiltonian for different $V_0/V_1$ for (a) $N=10$ and (b) $N=12$ (see Ref. 5 for plots of the $N=8$ case). No overlap data are available for the JSS for $N=12$.