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 ${\cal P}_{\rm LLL} \Phi_n\Phi_n^*\Phi_1^3$, where $\Phi_n$ is the wave function of the integer quantum Hall state with $n$ filled Landau levels and ${\cal P}_{\rm LLL}$ 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 $z_j=x_j-iy_j$ 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.
Highlights
The system of interacting electrons in the lowest Landau level (LLL) [or, in general, a given LL] has given rise to some of the most profound emergent structures found in nature
The first clue came from the observation of fractional quantum Hall effect (FQHE) at filling factor ν = 1/3 [1] in the lowest Landau level (LLL)
A large number of additional states were observed, primarily along the sequences ν = n/(2pn ± 1) (n and p are positive integers) and their hole conjugates ν = 1 − n/(2pn ± 1). These are understood as the integer quantum Hall effect (IQHE) of composite fermions (CFs) [4,5], which are bound states of electrons and an even number (2p) of quantized vortices
Summary
The system of interacting electrons in the lowest Landau level (LLL) [or, in general, a given LL] has given rise to some of the most profound emergent structures found in nature. The existence of charge e/(3n) excitations implies that the addition of a single flux quantum to the nn111 state produces n quasiholes This has implications for the structure of the low-energy spectrum slightly away from 7/3, which may explain the unusual nature of the exact Coulomb spectra slightly away from 7/3. The shift determines the Hall viscosity of a fractional quantum Hall state as ηH = hS/(24π 2 ), and the chiral central charge is related to the thermal Hall conductance as κxy = c− [π 2kB2 /(3h)]T , where T is the temperature The values of these topological quantities are given for the Laughlin state and for the particle-hole conjugate of the four-cluster Read-Rezayi (aRR4) state. VI with a discussion on the experimental implications of our results and an outlook for the future
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