Spinning Braid Group Representation and the Fractional Quantum Hall Effect

TL;DR

This paper generalizes the braid group representation to particles with spin using a path integral approach, leading to exact solutions for fractional quantum Hall states and insights into nonabelian anyons.

Contribution

It introduces a spin-inclusive braid group framework with charged winding numbers, enabling exact FQHE ground states and energy gap calculations.

Findings

Exact Laughlin, Halperin, and Moore-Read states derived

Energy gaps for quasi-excitations computed

Framework suggests spinning nonabelian anyons

Abstract

The path integral approach to representing braid group is generalized for particles with spin. Introducing the notion of {\em charged} winding number in the super-plane, we represent the braid group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the Hamiltonian, suggesting the possibility of {\em spinning nonabelian} anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as {\em exact} ground state solutions to the respective Hamiltonians associated to the braid group representations. The energy gap of the quasi-excitation is also obtainable from this approach.