Molecular anyons in fractional quantum Hall effect

Abstract

One of the profound consequences of the fractional quantum Hall (FQH) effect is the notion of fractionally charged anyons. In spite of extensive experimental study, puzzles remain, however. For example, both shot-noise and Aharonov-Bohm interference measurements sometimes report a charge that is a multiple of the elementary charge. We report here high-precision microscopic calculations that reveal the surprising result that the FQH anyons often bind together into stable clusters, which we term molecular anyons. This is counterintuitive, given that the elementary anyons carry the same charge and are therefore expected to repel one another. The number of anyons in a cluster, its binding energy and its size depend sensitively on the parent FQH state and the interaction between electrons (which is experimentally tunable, e.g., by varying the quantum well width). Our calculations further suggest that the charge-$1/4$ non-Abelian anyons of the $5/2$ FQH state may also bind to form charge-$1/2$ Abelian clusters. The existence of molecular anyons not only can provide a natural explanation for the observed charges, but also leads to a host of new predictions for future experiments and invites a re-analysis of many past ones.

Figures (4)

• Figure 1: (a) Pictorial depiction of direct tunneling from one edge to another (both edges shown as vertical lines) across a constriction, and cooperative tunneling through a localized molecular anyon. The molecular anyon shown here contains two QPs. (b) The panels show the structure of molecular anyons in terms of CF occupation of $\Lambda$ levels; a CF is depicted as an electron with two arrows. $\mathrm{QP}_{q}$ consists of an integer number of fully occupied $\Lambda$ levels and additional $q$ CFs in the lowest empty $\Lambda$ level. $\mathrm{QH}_{q}$ has an integer number of $\Lambda$ level with $q$ missing CFs from the topmost occupied $\Lambda$ level. (c) The spatial density profiles $\Delta \nu(x,y)/\nu$ of the lowest energy molecular anyons $\mathrm{QH}_{q}$ and $\mathrm{QP}_{q}$ pinned at the origin for the Jain fractions $\nu=2/5$, $3/7$, and the even-denominator state $\nu=5/2$. The magnetic length is denoted by $\ell$.
• Figure 2: The binding energy $\Delta_q$ (in units of $e^{2}/\varepsilon \ell$) of the molecular anyons $\mathrm{QP}_{q}$ and $\mathrm{QH}_{q}$ at fillings $\nu = 2/5$ and $3/7$ as a function of $1/N$, where $N$ is the number of electrons in the system. The results are for a pure 2D system. A negative intercept in the limit $N^{-1}\rightarrow 0$ indicates stability of the molecule.
• Figure 3: Stability regions for $\mathrm{QP}_{q}$ molecules at $\nu = 2/5$ and $\mathrm{QP}_{q}$ and $\mathrm{QH}_{q}$ molecules at $\nu = 3/7$, plotted as a function of quantum well width $w$ and electron density $\rho$. These results are based on systems with $N \sim 120$, where particles are considered bound if the probability that $\Delta_{q}$ is negative, as estimated from Monte Carlo simulations, exceeds $90\%$. The boundaries are accurate to within $\Delta w = \pm 5\;\mathrm{nm}$ and $\Delta \rho = \pm 1.25 \times 10^{10}\;\mathrm{cm}^{-2}$. At $\nu = 2/5$, only $\mathrm{QH}_{1}$ is stabilized.
• Figure 4: Binding energy $\Delta_{2}$ of two QHs of the $\nu=1/2$ non-Abelian parton state $\bar{2}\bar{2}111$, obtained at each relative angular momentum $L_{\mathrm{rel}}$ by CF diagonalization. The energies are measured relative to the state with $L_{\mathrm{rel}} \rightarrow \infty$, which represents far separated QHs. The upper panel considers QHs in distinct parton sectors, i.e. in different $\Phi_2$ factors, referred to as "topological"; this requires odd $N$. The lower panel considers QHs in the same parton sector, i.e. in the same $\Phi_2$ factor, referred to as "non-topological"; this corresponds to even $N$.