Monte Carlo Evaluation of Non-Abelian Statistics

We develop a general framework to (numerically) study adiabatic braiding of quasiholes in fractional quantum Hall systems. Specifically, we investigate the Moore-Read (MR) state at $\nu =1/2$ filling factor, a known candidate for non-Abelian statistics, which appears to actually occur in nature. The non-Abelian statistics of MR quasiholes is demonstrated explicitly for the first time, confirming the results predicted by conformal field theories.

Figure 1

The Berry phase $\chi$ for looping one MR quasihole around the equator with another quasihole fixed at a zenith angle $\xi$. $N=4,8,16,32,64$ is the number of electrons. The dashed line, $\chi /\pi =-1/8$, shows a naive prediction. For $\cos \xi \approx 0$, the two quasiholes approach each other very closely and we see strong finite-size oscillations in the Berry phase. For larger $N$ and $\cos \xi$ (i.e., larger quasihole separation in units of the magnetic length), $\chi$ appears to be converging toward zero. $\chi (-\cos \xi )=-\chi (\cos \xi )$.

Figure 2

For $\chi >0$ ($\chi <0$) filled symbols show the phase accumulated by interchanging two quasiholes around a circle with opening angle $\xi$ centered on the equator (north pole), for various $N$ as in Fig. 1. The straight dashed lines in the upper half are $0.5(N+1/4)(1-\cos \xi )$, corresponding to the expectation of the number of electrons enclosed by the loop. The $+1/4$ accounts for the charge pushed out by one of the quasiholes. For $\chi <0$, the dashed lines are $-0.5(N+1/4)\cos \xi$, i.e., one-half of the number of electrons inside minus one-half the number outside the loop. Open symbols, corresponding to a similar calculation with one quasihole moving and the other fixed at the center of the circle, almost overlay the filled symbols, confirming the trivial relative statistics.

Figure 3

Same as the upper half of Fig. 2, but now with four quasiholes present, two of which are fixed on the equator, at $\varphi =\pm 3\pi /4$, and two interchanged, with initial and final positions at $\varphi =\pm \xi$ on the equator. The straight dashed lines are $0.5(N+3/4)\cos \xi -1/4$. Here, $+3/4$ accounts for the average electron-density correction for the charge localized at $2n-1$ quasiholes. The additional phase offset of $-1/4$ reflects the Abelian part of the braiding statistics, in agreement with the predictions of Refs. 3, 10.

Figure 4

Parameters $\eta$ and $\beta$ defining transformation matrix (5) for the same operations as $\chi$ shown in Fig. 3. The dashed line shows $1/4$, an approximation used for $\eta$ in the text. Similarly $\beta$ can be approximated as zero [so that $ϵ$ in Eq. (5) is not defined]. These approximations become better with larger system size and for intermediate $\cos \xi$ when the quasiholes remain farther apart. The symbol convention is the same as in Fig. 3. Lines interpolate Monte Carlo results.