Spin-statistics relation for quantum Hall states

We prove a generic spin-statistics relation for the fractional quasiparticles that appear in Abelian quantum Hall states on the disk. The proof is based on an efficient way for computing the Berry phase acquired by a generic quasiparticle translated in the plane along a circular path, and on the crucial fact that once the gauge-invariant generator of rotations is projected onto a Landau level, it fractionalizes among the quasiparticles and the edge. Using these results we define a measurable quasiparticle fractional spin that satisfies the spin-statistics relation. As an application, we predict the value of the spin of the composite-fermion quasielectron proposed by Jain; our numerical simulations agree with that value. We also show that Laughlin's quasielectrons satisfy the spin-statistics relation, but carry the wrong spin to be the antianyons of Laughlin's quasiholes. We continue by highlighting the fact that the statistical angle between two quasiparticles can be obtained by measuring the angular momentum while merging the two quasiparticles. Finally, we show that our arguments carry over to the non-Abelian case by discussing explicitly the Moore-Read wave function.

Figure 1

Action of $\tilde{L}$. (a) Contour plot in $z$ space of the function $f(z,\eta )=exp[-\frac{1}{2}\text{Re}{(z-\eta )}^2-2\text{Im}{(z-\eta )}^2]$ for $\eta =2$. (b) Contour plot of $f(z{e}^{i\beta },\eta )$ for $\beta =2\pi /3$: With respect to (a), the plot is translated and rotated. (c) Contour plot of $f(z,\eta e^{i\beta })$: This time, the plot is only translated. (d) Contour plot of $f(z{e}^{i\beta },\eta e^{i\beta })$: The composition of the two is just a rotation and thus $\tilde{L}$ generates the self-rotations of the quasiparticles in the $z$ plane.

Figure 2

Calculation of the QE spin via the integral $J\left(r\right)={\int }_0^r(\frac{|r^{\prime }{|}^2}{2{\ell }_B^2}-1){\rho }_{\text{qp}}\left(r^{\prime }\right)2\pi r^{\prime }d{r}^{\prime }$; the spin of Eq. (7) coincides with the plateau appearing when $r$ is far from the center and the boundary; $R_{\mathrm{cl}}=\sqrt{2N/\nu }$ is the classical radius of the droplet. (a) The spin of a single Jain's QE for $\nu =1/2$, $1/3$, and $1/4$. (b) The case of two stacked Jain's QEs. (c) and (d) The same for one and two Laughlin's QEs, respectively. Theoretical predictions following from the SSR relation in Table 1 are marked with dashed lines and are only compatible with the spin of Jain's QE. Dashed-dotted lines in (c) and (d), together with their values, highlight the position of the spin plateau for Laughlin's QE.

Figure 3

Angular momentum $L(R_1,R_2)$ of a Laughlin state ($N=25$, $\nu =1/2$) with two QHs at distances $R_1$ and $R_2$ from the center, computed with Monte Carlo techniques [47, 48]. (a) Displacement of the first QH; the angular momentum variation $L(R,R_0)-L(R_0,R_0)$ is plotted in black circles, and it is a quadratic function of $R$ that agrees with the theory prediction $-\epsilon (R^2-R_0^2)$ (red line). (b) Displacement of the second QH; the variation $L(0,R)-L(0,R_0)$ is plotted in brown triangles and it is a quadratic function of $R$ only at large $R$; when the QHs fuse a deviation sets in that equals $-\kappa$, the statistical parameter.

Figure 4

Comparison of the quasihole spins $J\left(r\right)={\int }_0^r(\frac{{|r^{\prime }|}^2}{2{\ell }_B^2}-1){\rho }_{\text{qp}}\left(r^{\prime }\right)2\pi r^{\prime }d{r}^{\prime }$, for the different Moore-Read quasiholes: (a) the $(\sigma ,\frac{1}{2q})$, (b) the $(\mathbf{1},\frac{1}{q})$, and (c) the $(\psi ,\frac{1}{q})$, for the bosonic filling $\nu =1$ (corresponding to $q=1$) and the fermionic $\nu =\frac{1}{2}$ ($q=2$). $R_{\mathrm{cl}}=\sqrt{2N/\nu }$ is the classical radius of the droplet and the number of particles is $N=200$.