Ultimate braid-group generators for coordinate exchanges of Ising anyons from the multi-anyon Pfaffian wave functions

TL;DR

This paper rigorously derives the braid matrices for exchanging Ising anyons in Pfaffian quantum Hall states, establishing their equivalence to spinor representations of SO(2n) and confirming a key conjecture in topological quantum computation.

Contribution

It provides explicit formulas and recursive relations for braid group generators of Ising anyons, and proves their equivalence to SO(2n) spinor representations, advancing understanding of anyon braiding.

Findings

Explicit braid matrices for Ising anyons derived

Recursive relations for braid group generators established

Proved equivalence to SO(2n) spinor representations

Abstract

We give a rigorous and self-consistent derivation of the elementary braid matrices representing the exchanges of adjacent Ising anyons in the two inequivalent representations of the Pfaffian quantum Hall states with even and odd number of Majorana fermions. To this end we use the distinct operator product expansions of the chiral spin fields in the Neveu-Schwarz and Ramond sectors of the two-dimensional Ising conformal field theory. We find recursive relations for the generators of the irreducible representations of the braid group B_{2n+2} in terms of those for B_{2n}, as well as explicit formulas for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wave functions are completely equivalent to the spinor representations of SO(2n+2) and give the equivalence matrices explicitly. This actually proves that the correlation functions of 2n chiral Ising spin fields sigma do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2n) as conjectured by Nayak and Wilczek.