Measurable fractional spin for quantum Hall quasiparticles on the disk

We study the spin of the localized quasiparticle excitations of lowest-Landau-level quantum Hall states defined on a disk. The spin that we propose satisfies the spin-statistics relation and can be used to reconstruct the topological geometric phase associated to the exchange of two arbitrarily chosen quasiparticles. Since it is related to the quadrupole moment of the quasiparticle charge distribution, it can be measured in an experiment and could reveal anyonic properties in a way that is complementary to the interferometric schemes employed so far. We first discuss our definition for the quasiholes of the Laughlin state, for which we present a numerical and analytical study of our spin, and we proceed with a discussion of several kinds of quasiholes of the Halperin 221 state. Finally, we discuss the link between our spin and the adiabatic rotation of the quasiparticles around their axis and demonstrate that our spin obeys the spin-statistics relation.

Figure 1

Fractional spin of $q$ quasiholes of a Laughlin state. Panels in first (second) row show $J_{m,q}\left(R\right)$ for $q=1$ ($q=2$), computed via Monte Carlo sampling and Eq. (6) (blue-solid lines), for Laughlin states with filling $1/m$ (see column titles) and $N=200$. Horizontal-dashed lines represent the saturation values $J_{m,q}$ reported in Table 1. Error bars are statistical uncertainties, and data are only shown for cutoff radii $R$ where boundary effects are negligible and Monte Carlo sampling is reliable.

Figure 2

Fractional spin of quasiholes of the Halperin 221 state, $J_{\mathrm{\Gamma },q}\left(R\right)$ for $q=1$ ($q=2$), as computed via Monte Carlo sampling and Eq. (6), for the Halperin 221 states with $N_A=N_B=150$ particles in each layer and for selected values of $(\mathrm{\Gamma },q)$. Note that some properties in Table 2 (e.g., for the exchange $A\leftrightarrow B$) are obtained by a combination of $J_{A,1}$ and $J_{AB,1}$, or by the fact that $J_{A,q}=J_{B,q}$. The bulk density reads $\overline{\rho }=(2/3)/\left(2\pi {\ell }_B^2\right)$. Error bars are statistical uncertainties, and data are only shown in a range of cutoff radii $R$ where boundary effects are negligible and Monte Carlo sampling is reliable.

Figure 3

(Left) Sketch of a FQHE state defined on a disk with two quasiholes located at ${\eta }_{\mu }$, $\mu =1,2$. The red lines indicate the trajectories of the quasiholes, which are rigidly rotated of an angle ${\theta }_f=2\pi$ at fixed distance $|{\eta }_i|$ from the center of the disk ($\times$). (Center and right) Two different ways of moving quasiparticles: without self-rotation, as in the left panel, or with self-rotation, as in the right panel. The former approach is the standard one, while the latter is the one considered in this article; in order to study it, we keep the quasiparticles fixed in the reference frame that rotates at angular frequency ${\vartheta }_f/T$.

Figure 4

FQHE state with one quasiparticle located at $\eta$. Red: Adiabatic motion of one quasiparticle around a circumference with radius $\left|\eta \right|$. Black: region $\mathcal{A}$ used to perform the integral in Eq. (51) and to avoid boundary effects.

Figure 5

LLL state with two quasiparticles located at ${\eta }_1$ and ${\eta }_2$. Red: Adiabatic motion of one quasihole around a circumference with radius $|{\eta }_{\mu }|$.

Figure 6

Laughlin density profile at the boundary, obtained by Monte Carlo sampling: (a) Density profiles for different sizes $N$, in the absence of quasiholes ($q=0$); (c) same data as in (a), after subtracting the curve for $N=300$; (b) density profiles for different numbers $q$ of quasiholes in the origin (for $N=100$); (d) same data as in (b), after subtracting the curve for $q=0$.

Figure 7

(a) Halperin 221 density profile at the boundaries for different numbers $q$ of $\mathrm{\Gamma }=A$ quasiholes in the origin (for $N=100$), after subtracting the $q=0$ curve; (b) same as in (a), for $\mathrm{\Gamma }=AB$.

Figure 8

Depletion density profile on the A and B layers, obtained by Monte Carlo sampling of the Halperin 221 state with $\mathrm{\Gamma }=A,AB$ and $q=1,2$. The depletion profile on each layer is the difference between the corresponding density profile and the layer bulk density $(1/3)/\left(2\pi l_B^2\right)$. For $q=1$, we observe a full agreement with data from Ref. [46] (black-dashed lines, labeled JRL2019).

Figure 9

Depletion profiles for the Laughlin state with filling $1/m$, for quasiholes with $q=1,2$, as obtained by sampling ${\mathrm{\Psi }}_{m,q}$ or by Eq. (D1).