Understanding the Effects of Electronic Interaction on Braiding of Non-Abelian Anyons in Fractional Quantum Hall Phases

Non-Abelian anyons in fractional quantum Hall (FQH) systems are typically ground states of model Hamiltonians with three- or more-body interactions. On the other hand, electronic interactions in real systems involve two-body interactions. Many past studies into the effect of realistic interaction is limited to Coulomb interactions, whereas the effects of individual Haldane potentials, which form a basis for any two-body interaction are not fully emphasizes since the traditional Monte-Carlo integration using first-quantized wavefunctions is not applicable. Using a more modern understanding, we can now understand quasiholes as the fundamental degrees of freedom of a given FQH systems which interact by additional interactions on top of the model Hamiltonian. For example, the Moore-Read (MR) quasiholes are the fundamental degree of freedom in a system whose Hamiltonian is dominated by the three-body interaction, and any additional two-body interaction results in both their creation energy and effective interaction.

This understanding allows us to make many predictions about how non-Abelian anyons behave under realistic interaction. For the MR quasiholes, we predict a "parity flip" when the two-body interaction is short-ranged. The parity conservation in the MR state dictates that when fusing pairs of quasiholes, such that each pair of charge-$e/4$ anyon forms a charge-$e/2$ anyon, the parity (oddness or evenness) of the number $\psi$-anyon must be conserved within a given system. This means that in a system with $2n$ quasiholes, fusing a pair of quasihole affects the fusion outcomes of the remaining $2n-2$ quasiholes regardless of the quasihole separation. Since the fusion outcome of two quasiholes directly affects their braiding statistics, this long-range ``entanglement" directly affects the measurement of exchange statistics. In experiments, it is therefore important to account for all quasiholes in the systems in order to understand the outcome of a certain measurement.