Abelian and Non-Abelian Hall Liquids and Charge-Density Wave: Quantum Number Fractionalization in One and Two Dimensions

Previously, we have demonstrated that, on a torus, the Abelian quantum Hall liquid is adiabatically connected to a charge-density wave as the smaller dimension of the torus is varied. In this work, we extend this result to the non-Abelian bosonic Hall state. The outcome of these works is the realization that the paradigms of quantum number fractionalization in one dimension (polyacetylene) and two dimensions (fractional quantum Hall effect) are, in fact, equivalent.

Figure 1

The large $\kappa$ limits of the (a)–(c) ground state, (d)–(f) and (i),(j) 2-hole excited states, and (g),(h) 4-hole excited state. A vertical segment indicates the position of a charge $\Delta Q=1/2$ domain wall; see text.

Figure 2

Curves: The charge-density wave order parameter $\mathcal{O}$ of the ground states in the zero center-of-mass sector for $N=4$, 6, 8, and 10 particles. These states are adiabatically connected to the (20)- and (02) states in Figs. 1a, 1b. The ground state that is connected to the (11) state in Fig. 1c is free of symmetry breaking and, hence, has $\mathcal{O}=0$. The single curve labeled “gap” shows the energy gap for $N=10$ versus $\kappa$ for $\kappa >{\kappa }_{\mathrm{sd}}$. Gap data are scaled down by a factor of 4 for convenience. Inset: The quantity $\Gamma =1/\sqrt{-\log \mathcal{O}}$.