Figure 1

Statistical phase for winding one qh around another in the spherical geometry as defined in the text. Data are shown for the case of even total numbers of electrons for which the topological quantum number of the pair of qh’s is 1. In (a), we have plotted the statistical phase versus the chord distance $R$ between the two qh’s and fit the data using $\cos (a\frac{R}{l_0}+b)$ for the oscillatory part and a decay term that is either exponential, Gaussian or power law. We fit the data starting at $R=6.48l_0$ and find the value of the reduced ${\chi }^2$ is smallest for exponential decay with a value of 1.42 while for power law and Gaussian decays, it is 7.22 and 5.52, respectively. The good fit to exponential decay is also confirmed when we plot the absolute value of the data on log and log-log scales (b) and perform linear fits to the extrema of the oscillations. The linear fit is clearly much better on the log plot demonstrating that the oscillations decay exponentially rather than as a power law.Reuse & Permissions

Figure 2

Energy splitting of the eigenstates on a sphere with four qh’s as a function of system size. The four qh’s are placed on the corners of an equilateral tetrahedron. The data is fit with the function $\cos (a\sqrt{N}+b)$ multiplied by an exponential, Gaussian, or power law decay function. The distance between particles grows as $\sqrt{N}$, so this is essentially the same fit as used in Fig. 1. The data is fit starting with $N=12$ (not shown because the fits are nearly identical at the small $N$ values). The reduced ${\chi }^2$ values for the exponential, Gaussian, and power law fits are 3.39, 7.73, and 6.49, respectively, which helps confirm our expectation that two energies become exponentially degenerate as the qh’s move apart. We have not shown the log and log-log plots here because such plots discard the sign, and in the absence of a higher density of points, are hard to interpret.Reuse & Permissions

Figure 3

This figure shows the total energy of the $N=40$ system with four qh’s as qh 2 is moved closer to its pair, qh 1. qh 1 is located at the north pole, and qh 2 is moved along a path that keeps a certain analytic form [8] for the trial states $|1⟩$ and $|\psi ⟩$ precisely orthogonal [17]. The two states of the system, $|1⟩$ and $|\psi ⟩$, which are nearly degenerate when the qh’s are well separated, split as the qh’s approach each other. The inset shows the energy needed to move qh 2 within ${\ell }_0$ of qh 1 for different values of $N$. We find that it takes more energy to bring the two qh’s together when the system is in state $|1⟩$ than in state $|\psi ⟩$, and that the energy splitting is on the order of $0.01e^2/ϵ{\ell }_0$.Reuse & Permissions