Shape Deformation and Braid Statistics of Fractional Quantum Hall Quasiparticles

For ``ideal anyons,'' the Berry phase associated with a closed loop of an anyon around another is robust, i.e., independent of the size and shape of the loop, and directly yields the braid statistics. This is not the case for fractional quantum Hall (FQH) quasiparticles (QPs), which are charged and have finite size. The Berry phase Θ associated with a loop of a fractionally charged QP around another contains two contributions: Θ=Θ AB+Θ braid, where Θ AB, the Aharonov-Bohm phase, is proportional to the enclosed area, and Θ braid is the order-one contribution from braid statistics. The latter is obtained by determining Θ for a closed loop twice, once with and once without the other QP inside, and taking the difference. It is known that even small uncertainties in the enclosed areas can produce corrections to Θ AB larger than Θ braid. We consider how the Berry phase depends on the shape of the QP, which, unlike its charge, is not topological and varies along the path in response to the local potential. Our calculations show that practically imperceptible differences in the shape of the outer QP in the two experiments can produce a non-universal value for the braid statistics, revealing an additional challenge for a precise determination of the braid statistics. We discuss these results in the context of interference experiments. We also note that the fractional phase jumps in these experiments can be understood without assuming QPs with sharply quantized fractional charges at the edges of the FQH system; instead, they directly measure the fractionally quantized vorticity of bulk QPs.