Long-range entanglement and role of realistic interaction in braiding of non-Abelian quasiholes in fractional quantum Hall phases

Parity conservation dictates that when fusing pairs of Moore-Read (MR) quasiholes, such that each pair of charge-$e/4$ anyons forms a charge-$e/2$ anyon, the parity of the numbers of 1-anyon and $\ensuremath{\psi}$-anyon must be conserved within a given system. This idea is illustrated here using the Jack polynomial formalism, which also provides a basis to numerically study the dynamics of MR anyons. We find that parity conservation gives rise to a long-range ``entanglement'' between statistics of one pair of anyons and the fusion channels of other anyons in the background. It is, therefore, important to account for all quasiholes in an experimental system in order to accurately predict the measurement outcome. We also examine the effect of two-body electron-electron interaction on the degeneracy of two anyon fusion channels. This understanding of the quasihole dynamics can help to fine-tune two-body interactions in order to stabilize the non-Abelian degeneracy, which is important for detecting non-Abelian braiding in experiments.