Skein Theory and Topological Quantum Registers: Braiding Matrices and Topological Entanglement Entropy of Non-Abelian Quantum Hall States

TL;DR

This paper uses skein theory to analyze non-Abelian quantum Hall states, computing braiding matrices and topological entanglement entropy, revealing insights into their topological properties and quantum dimensions.

Contribution

It introduces a skein-theoretic method to compute braiding matrices and entanglement entropy for non-Abelian quantum Hall states, generalizing previous models.

Findings

Computed braiding matrices for arbitrary spins in Read--Rezayi states.

Proposed skein-theoretic method to calculate topological entanglement entropy.

Identified the dependence of entanglement entropy on quantum dimensions.

Abstract

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read--Rezayi state whose effective theory is the SU(2)_K Chern--Simons theory. As a generalization of the Pfaffian (K=2) and the Fibonacci (K=3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.