Berry phases for composite fermions: Effective magnetic field and fractional statistics

The quantum Hall superfluid is presently the only viable candidate for a realization of quasiparticles with fractional Berry phase statistics. For a simple vortex excitation, relevant for a subset of fractional Hall states considered by Laughlin, nontrivial Berry phase statistics were demonstrated many years ago by Arovas, Schrieffer, and Wilczek. The quasiparticles are in general more complicated, described accurately in terms of excited composite fermions. We use the method developed by Kjønsberg, Myrheim, and Leinaas to compute the Berry phase for a single composite-fermion quasiparticle and find that it agrees with the effective magnetic field concept for composite fermions. We then evaluate the “fractional statistics,” related to the change in the Berry phase for a closed loop caused by the insertion of another composite-fermion quasiparticle in the interior. Our results support the general validity of fractional statistics in the quantum Hall superfluid, while also giving a quantitative account of corrections to it when the quasiparticle wave functions overlap. Many caveats, both practical and conceptual, are mentioned that will be relevant to an experimental measurement of the fractional statistics. A short report on some parts of this article has appeared previously.

Figure 1

The logical path to fractional statistics. First, the interacting electrons transform into weakly interacting composite fermions at an effective magnetic field. Composite fermions form incompressible states at certain fractional fillings of the lowest Landau level. Incompressibility implies fractional charge, which, finally, leads to fractional statistics.

Figure 2

Excess charge density relative to the ground state in the presence of a single CFQP. We used $\nu =1∕3$, $N=50$, and $\eta ∕l=0.3R_d\approx 5.2$ with disk size $R_d\equiv \sqrt{2N∕\nu }$. The resulting position of the CFQP is in good agreement with the intended position, which is indicated by the arrow in the contour plot (lower panel).

Figure 3

The Berry phase ${\Phi }^{*}$ for a single CFQP at $\nu =1∕3$ (upper panel) and $\nu =2∕5$ (lower panel) as a function of $\eta$. Here $N$ is the total number of composite fermions, $l$ is the magnetic length, and ${\Phi }_e\equiv -2\pi BA∕{\varphi }_0$ is the Berry phase acquired by an electron moving in an empty space. The error bars from Monte Carlo sampling which are smaller than the symbol size are not shown explicitly. The deviation at the largest $\eta ∕l$ for each $N$ is due to proximity to the edge.

Figure 4

Actual location $\xi$ of a CFQP (in units of the disk size $R_d\equiv l\sqrt{2N∕\nu }$) as a function of the parameter $\eta$ for $\nu =1∕3$ and $N=50$ when the CFQP is outside the quantum Hall superfluid droplet. The dashed line is $\xi =\eta$ and the solid line is $\xi =\eta ∕(2pn+1)$, the position the CFQP would have in the absence of any other composite fermions (in the case of $\nu =1∕3$, we have $\xi =\eta ∕3$; see text).

Figure 5

Berry phase ${\Phi }^{*}$ acquired by a CFQP when it is outside the quantum Hall droplet. The system has $N=50$ composite fermions at the filling $\nu =1∕3$. The dashed line is the prediction of the CF theory, and the squares are calculated from the microscopic wave functions.

Figure 6

The statistical angle ${\tilde{\theta }}^{*}$ for the CFQP’s at $\nu =1∕3$ (upper panel) and $\nu =2∕5$ (lower panel) as a function of $d\equiv \mid \eta -{\eta }^{\prime }\mid$. Here $N$ is the total number of composite fermions, and $l$ is the magnetic length. The error bar from Monte Carlo sampling is not shown explicitly when it is smaller than the symbol size. The deviation at the largest $d∕l$ for each $N$ is due to proximity to the edge. This figure was shown earlier in Ref. 20 and is reproduced here for completeness.

Figure 7

The statistical angle ${\tilde{\theta }}^{*}$ for the CFQP’s at $\nu =1∕3$ for large $d\equiv \mid \eta -{\eta }^{\prime }\mid$. Here $N$ is the total number of composite fermions, and $l$ is the magnetic length. The dashed line is Eq. (56), which is in good agreement with the long tail of the numerical data. The points near the edge deviate significantly from the dashed line.

Figure 8

The statistical angle ${\tilde{\theta }}^{*}$ for the CFQP’s at $\nu =1∕3$ (left panel) and $\nu =2∕5$ (right panel) for small $d\equiv \mid \eta -{\eta }^{\prime }\mid$. The symbols are the same as in Fig. 7, and $l$ is the magnetic length. The heuristic formula in Eq. (60) (dashed lines) agrees well with the actual result for small $d$.

Figure 9

The statistical angle ${\tilde{\theta }}^{*}$ for the CFQP’s in different CF-quasi-LL’s at the filling $\nu =1∕3$ as a function of $d\equiv \mid \eta -{\eta }^{\prime }\mid$. The CFQP at the origin is in the third CF-quasi-LL while the CFQP traversing a closed loop is in the second CF-quasi-LL. Here $N$ is the total number of composite fermions, and $l$ is the magnetic length.