Long-range Entanglement and Role of Realistic Interaction in Braiding of Non-Abelian Quasiholes in Fractional Quantum Hall Phases

TL;DR

The paper identifies parity conservation in Moore-Read fractional quantum Hall states as a source of long-range entanglement that couples distant quasihole fusion channels, potentially altering braiding statistics in realistic two-body interacting systems. Using Jack polynomials to construct MR quasihole wavefunctions and exact diagonalization, the authors quantify how two-body pseudopotentials $V_m^{2bdy}$ shape the self-energies of $1$- and $\psi$-type quasiholes and, consequently, the fusion-channel energetics that govern braiding. They demonstrate that background fusion outcomes can deterministically affect the statistics of other quasiholes, and show how tuning the relative strengths of $V_1^{2bdy}$ and $V_3^{2bdy}$—through layer thickness, screening, or pinning—can minimize the non-Abelian gap and stabilize the desired degeneracy. These results provide practical guidance for realizing robust non-Abelian braiding in MR systems and offer a framework extendable to other non-Abelian fractional quantum Hall states.

Abstract

Parity conservation dictates that when fusing pairs of Moore-Read (MR) quasiholes, such that each pair of charge-$e/4$ anyon forms a charge-$e/2$ anyon, the parity of the numbers of $1$-anyon and $ψ$-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. In particular, we examine the effect of two-body electron-electron interaction on the degeneracy of two anyon fusion channels, which affects their mutual statistics of the MR anyons. We find that parity conservation gives rise to a long-range ``entanglement" which affect the experimental measurement of exchange statistics under realistic electron interaction. It is therefore important to account for all quasiholes in an experimental systems in order to accurately predict the outcome of a certain measurement. We also show how understanding the quasihole dynamics can help to fine-tune two-body interactions in order to stabilize any given fusion channel in experiments.

Figures (13)

• Figure 1: If in the Moore-Read (MR) phase both charge-$e/4$ (yellow circles) and charge-$e/2$ (fused pair of circles) are present, the anyonic species of the charge-$e/2$ quasiholes will be determined by the underlying electronic interaction in the system. This in turn will affect the braiding measurement via a long-range entanglement originating from parity conservation.
• Figure 2: (a) The spherical geometry used in this paperhaldane1983fractional: a magnetic monopole of strength $2S\hbar$ (where $S$ is either integer or half-integer) is placed at the center of a sphere, creating a uniform magnetic field everywhere perpendicular to the surface of the sphere. The Landau orbitals are concentric rings parallel to the equatorgreiter2011landau(b) A root configuration is a binary string with each digit representing an orbital from the north pole to the south pole going right to left. To read off the position of MR quasiholes, every group of four consecutive digits is checked (examplified by yellow and purple ellipses) and a quasihole is marked whenever there are fewer than two electrons in it (examplified by yellow ellipses). The root is extended on the two sides by "virtual" repetitions of "1100" in order to read off the quasiholes residing at the two poles. (c) The density per orbital on the sphere showing the three types of anyons in the Moore-Read state: two $\sigma$-anyons at each pole (blue); one $1$-type at the north pole (orange); and one $\psi$-type at the north pole (green).
• Figure 3: Finite-size scaling of the overlap of the two Jacks describing different types of MR 2-stack quasiholes. Their roots are given by Eq.(\ref{['two 1-types']}) and Eq.(\ref{['two psi-types']}). Maximum system size used in calculation has $N_e=18$ electrons.
• Figure 4: If the local measurements (dashed box) do not account for all quasiholes in the system, the behavior of the local system may differ from the global one depending on the effective electron-electron interaction (a) Illustration of the degeneracy-splitting mechanism (b) Relation of the parity of the low-lying states after the degeneracy splitting with that of the original system. Highlighted in red are the cases where the parity of the low-lying manifold is flipped after degeneracy splitting.
• Figure 5: Difference in self-energy of the two types ($E_1-E_\psi$) with respect to $\hat{V}_{m}^{2bdy}$ for $m=1$ (blue circles), $m=3$ (red stars), $m=5$ (yellow diamons), $m=7$ (purple squares), and m=9 (green crosses). Maximum system size used in calculation has $N_e=18$ electrons. A positive energy difference implies that the $\psi$-type quasihole is energetically favorable compared to $1$-type, and vice versarawdata.