$2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States

TL;DR

This paper demonstrates that in paired quantum Hall states, the $2n$ quasihole wavefunctions exhibit an exponential degeneracy, realizing a spinor representation of a nonabelian statistics group, with implications for quantum statistics and potential glassy behavior.

Contribution

It explicitly constructs a basis for quasihole states in the $ u=1/2$ Pfaffian state, revealing their realization of a spinor representation of $SO(2n)$ and supporting the universality of nonabelian statistics.

Findings

Exponential degeneracy of quasihole states at fixed positions.

Explicit calculation of conformal blocks for four quasiholes.

Prediction of glassy behavior in the system.

Abstract

By explicitly identifying a basis valid for any number of electrons, we demonstrate that simple multi-quasihole wavefunctions for the $\nu=1/2$ Pfaffian paired Hall state exhibit an exponential degeneracy at fixed positions. Indeed, we conjecture that for $2n$ quasiholes the states realize a spinor representation of an expanded (continuous) nonabelian statistics group $SO(2n)$. In the four quasihole case, this is supported by an explicit calculation of the corresponding conformal blocks in the $c={1\over2}+1$ conformal field theory. We present an argument for the universality of this result, which is significant for the foundations of fractional statistics generally. We note, for annular geometry, an amusing analogue to black hole entropy. We predict, as a generic consequence, glassy behavior. Many of our considerations also apply to a form of the (3,3,1) state.