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

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 numbers 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 in terms of those for , as well as explicit formulae for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wavefunctions 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 σ do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2n) as conjectured by Nayak and Wilczek.