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, that is, independent of the size or the shape of the loop, and directly yields the braid statistics. That is not the case for the fractional quantum Hall (FQH) quasiparticles (QPs), which are charged and have finite size. We consider here how the Berry phase depends on the shape of the QP, which, unlike its charge, is not a topological property and varies along the path in response to the local potential. We show that the Berry phase $Θ$ associated with the loop of a fractionally charged QP around another contains three distinct contributions: $Θ=Θ_{\rm AB}+Θ_{\rm shape}+Θ_{\rm braid}$. The Aharonov-Bohm phase $Θ_{\rm AB}$ is dominant, being proportional to the area of the loop, and the shape-dependent term $Θ_{\rm shape}$, identified in this work, can be larger than the order-one contribution from the braid statistics $Θ_{\rm braid}$. A precise determination of the braid statistics is challenging because it can be swamped by practically undetectable uncertainties in its trajectory and shape. We discuss these results in the context of the interference experiments. We also note that the fractional phase jumps in these experiments can be understood without assuming the existence of QPs with sharply quantized fractional charges at the edges of the FQH system, wherein these phase jumps are a direct measure of the fractionally quantized vorticity of the QPs in the bulk of the fractional quantum Hall state.