Topological degeneracy of non-Abelian states for dummies

TL;DR

This paper constructs and analyzes the degenerate groundstates of non-Abelian Moore-Read Pfaffian states on Riemann surfaces, revealing their topological properties and degeneracies, and relates them to p+ip superconductors.

Contribution

It provides a physical construction of non-Abelian groundstates on arbitrary Riemann surfaces, generalizing previous geometric approaches and clarifying the role of fractional charge.

Findings

Reproduces known groundstate degeneracy exactly

Discusses restrictions on statistics due to fractional charge

Connects Pfaffian states with p+ip superconductors

Abstract

We present a physical construction of degenerate groundstates of the Moore-Read Pfaffian states, which exhibits non-Abelian statistics, on general Riemann surface with genus g. The construction is given by a generalization of the recent argument [M.O. and T. Senthil, Phys. Rev. Lett. 96, 060601 (2006)] which relates fraction- alization and topological order. The nontrivial groundstate degeneracy obtained by Read and Green [Phys. Rev. B 61, 10267 (2000)] based on differential geometry is reproduced exactly. Some restrictions on the statistics, due to the fractional charge of the quasiparticle are also discussed. Furthermore, the groundstate degeneracy of the p+ip superconductor in two dimensions, which is closely related to the Pfaffian states, is discussed with a similar construction.