Bridge Between Abelian and Non-Abelian Fractional Quantum Hall States

We propose a scheme to construct the most prominent Abelian and non-Abelian fractional quantum Hall states from $K$-component Halperin wave functions. In order to account for a one-component quantum Hall system, these $\mathrm{SU}(K)$ colors are distributed over all particles by an appropriate symmetrization. Numerical calculations corroborate the picture that $K$-component Halperin wave functions may be a common basis for both Abelian and non-Abelian trial wave functions in the study of one-component quantum Hall systems.

Figure 1

Thin-torus limit of ${\mathcal{S}}^{\prime }{\Psi }_{[1;3]}^{(K)}$ for different values of $K$ and $N$. The particles phase separate into $K$ repelling droplets, as expected for $m<n$. In general, these highest weights and the “squeezing” technique [15] lead neither to a unique quantum state nor to a state identical to that obtained by color symmetrization.

Figure 2

Thin-torus limit of ${\mathcal{S}}^{\prime }{\Psi }_{[m;n]}^{(K)}$. The shaded boxes indicate clusters of $K$ particles. The spacing between the clusters is determined by the exponent $m$, whereas that within each cluster by $n$, i.e., the intercolor correlations. $n=0$ (for bosons) corresponds to particles on the same site within each cluster.