${\mathbb{Z}}_n$ superconductivity of composite bosons and the $7/3$ fractional quantum Hall effect

Abstract

The topological $p$-wave pairing of composite fermions, believed to be responsible for the 5/2 fractional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent "superconductivity" in the 1/3 FQHE, which involves condensation of clusters of $n$ composite bosons. From a microscopic perspective, the state is described by the $n\bar{n}111$ parton wave function ${\mathcal{P}}_{\mathrm{LLL}}{\Phi }_n{\Phi }_n^{\ast }{\Phi }_1^3$, where ${\Phi }_n$ is the wave function of the integer quantum Hall state with $n$ filled Landau levels and ${\mathcal{P}}_{\mathrm{LLL}}$ is the lowest-Landau-level projection operator. It represents a ${\mathbb{Z}}_n$ superconductor of composite bosons, because the factor ${\Phi }_1^3\sim {\prod }_{j<k}(z_j-z_k{)}^3$, where $z_j=x_j-i{y}_j$ is the coordinate of the $j$th electron, binds three vortices to electrons to convert them into composite bosons, which then condense into the ${\mathbb{Z}}_n$ superconducting state $|{\Phi }_n{|}^2$. From a field theoretical perspective, this state can be understood by starting with the usual Laughlin theory and gauging a ${\mathbb{Z}}_n$ subgroup of the $U(1)$ charge conservation symmetry. We find from detailed quantitative calculations that the $2\bar{2}111$ and $3\bar{3}111$ states are at least as plausible as the Laughlin wave function for the exact Coulomb ground state at filling $\nu =7/3$, suggesting that this physics is possibly relevant for the 7/3 FQHE. The ${\mathbb{Z}}_n$ order leads to several observable consequences, including quasiparticles with fractionally quantized charges of magnitude $e/(3n)$ and the existence of multiple neutral collective modes. It is interesting that the FQHE may be a promising venue for the realization of exotic ${\mathbb{Z}}_n$ superconductivity.

17 pages, 7 figures, published version