Cyclic Statistics In Three Dimensions

TL;DR

This paper demonstrates the existence of cyclic, non-permutation group statistics in three-dimensional quantum systems of unknotted rings, challenging the traditional view that 3D quantum statistics are solely permutation-based.

Contribution

It provides the first explicit example of cyclic, Z_n, non-permutation group statistics in 3D systems, using topological and algebraic methods.

Findings

Existence of cyclic Z_n statistics in 3D systems of rings

Use of Goldsmith's theorem and Fuchs-Rabinovitch relations

Challenging the assumption that 3D quantum statistics are permutation-only

Abstract

While 2-dimensional quantum systems are known to exhibit non-permutation, braid group statistics, it is widely expected that quantum statistics in 3-dimensions is solely determined by representations of the permutation group. This expectation is false for certain 3-dimensional systems, as was shown by the authors of ref. [1,2,3]. In this work we demonstrate the existence of ``cyclic'', or $Z_n$, {\it non-permutation group} statistics for a system of n > 2 identical, unknotted rings embedded in $R^3$. We make crucial use of a theorem due to Goldsmith in conjunction with the so called Fuchs-Rabinovitch relations for the automorphisms of the free product group on n elements.