Numerical demonstration of Abelian fractional statistics of composite fermions in the spherical geometry

TL;DR

This paper numerically demonstrates Abelian fractional statistics for composite-fermion particle excitations in Jain FQH states using the spherical (Haldane) geometry. By constructing CFPs via coherent states and performing controlled braiding on the sphere, the authors extract the statistics parameter $\Theta=\tfrac{2p}{2pn+1}$ for fractions $\nu=n/(2pn+1)$ and show that sphere curvature removes the edge- or shift-related ambiguities present in disk geometries. The results for $\nu=1/3,1/5,2/5,2/9,3/7$ corroborate theoretical predictions with accurate convergence at modest system sizes (e.g., $N\sim50$ for the simplest states). The methodology provides a robust framework for bulk braiding in finite systems and can be extended to non-Abelian parton states and related diagnostic approaches.

Abstract

Fractional quantum Hall (FQH) fluids host quasiparticle excitations that carry a fraction of the electronic charge. Moreover, in contrast to bosons and fermions that carry exchange statistics of $0$ and $π$ respectively, these quasiparticles of FQH fluids, when braided around one another, can accumulate a Berry phase, which is a fractional multiple of $π$. Deploying the spherical geometry, we numerically demonstrate that composite fermion particle (CFP) excitations in the Jain FQH states carry Abelian fractional statistics. Previously, the exchange statistics of CFPs were studied in the disk geometry, where the statistics get obscured due to a shift in the phase arising from the addition of another CFP, making its determination cumbersome without prior knowledge of the shift. We show that on the sphere this technical issue can be circumvented and the statistics of CFPs can be obtained more transparently. The ideas we present can be extended to determine the statistics of quasiparticles arising in certain non-Abelian partonic FQH states.

Figures (4)

• Figure 1: Pictorial depiction of the phase, $\xi^{*}$, accumulated when a CF particle (electron and two vortices depicted by a ball and two vertical arrows) loops on the Haldane sphere. Red arrows show flux emanating from the magnetic monopole of strength $2Q$ placed at the origin that coincides with the sphere's center. (a) A single CF particle moving along the latitude $\theta$ picks up an Aharonov-Bohm phase proportional to the area of the loop it encloses. (b) A CF particle moving along latitude $\theta$ in the presence of another CF particle static at the north pole picks up an additional phase (shown in red). The green dotted loop shows an example of a path wherein the singularities of the vector potential at the north and south poles are placed on the same side of the path.
• Figure 2: Berry phase accumulated when a CFP moves along the equator ($\theta{=}\pi/2$) with another CFP kept static at the north pole for Jain states at $\nu{=}n/(2pn{+}1)$. The expected phase (green line) is equal to ${-}\Theta{=}{-}2p{/}(2pn{+}1)$ and is plotted alongside the numerical result (red dot with error bar determined from the statistical uncertainty of the Monte Carlo simulation) for the largest system size corresponding to each fraction as listed in Table \ref{['tab: frac_stat']}.
• Figure 3: Berry phase accumulated by a single CFP looping along various latitudes making an angle $\theta$ with the $z$-axis for many filling fractions along the Jain sequence for three different system sizes. Dots and lines correspond to numerically computed phase and theoretically-predicted phases [see Eq. \ref{['eq: single qp phase']}] respectively.
• Figure 4: Phase accumulated due to a CFP looping along various latitudes $\theta$ in the presence of another CFP static at the north pole for multiple fillings. Solid dots and solid lines indicate numerical data and curve fitting using Eq. \ref{['eq: mod_doub_qp_phase_final']} respectively.