Localized quasiholes and the Majorana fermion in fractional quantum Hall state at nu=5/2 via direct diagonalization

TL;DR

This study uses exact diagonalization to analyze localized quasiholes in the fractional quantum Hall state at nu=5/2, demonstrating the localization of e/4 quasiholes and the potential existence of Majorana fermions, with implications for topological quantum computation.

Contribution

It provides a detailed numerical investigation of quasihole localization and adiabatic continuity in the nu=5/2 state, supporting the presence of Majorana fermions under Coulomb interactions.

Findings

e/4 quasiholes can be localized with a delta-function potential

Adiabatic continuity exists between Moore-Read and Coulomb states for two and four quasiholes

Preliminary braiding results suggest Majorana fermions in Coulomb limit

Abstract

Using exact diagonalization in the spherical geometry, we investigate systems of localized quasiholes at nu=5/2 for interactions interpolating between the pure Coulomb and the three-body interaction for which the Moore-Read state is the exact ground state. We show that the charge e/4 quasihole can be easily localized by means of a delta-function pinning potential. Using a tuned smooth pinning potential, the quasihole radius can be limited to approximately three magnetic length units. For systems of two quasiholes, adiabatic continuity between the Moore-Read and the Coulomb limit holds for the ground state, while for four quasiholes, the lowest two energy states exhibit adiabatic continuity. This implies the existence of a Majorana fermion for pure Coulomb interaction. We also present preliminary results in the Coulomb limit for braiding in systems containing four quasiholes, with up to 14 electrons, diagonalizing in the full spin-polarized sector of the second Landau-level Hilbert space.