Anyons in One Dimension

TL;DR

This paper provides a non-technical overview of fractional statistics in one-dimensional systems, highlighting how anyons acquire fractional phases and shifts in momentum spacings under periodic boundary conditions.

Contribution

It introduces the concept of fractional phase shifts as good quantum numbers for interacting anyons in one dimension, despite non-integer momentum quantization.

Findings

Crossing of anyons is uni-directional in 1D systems.

Fractional phase $ heta$ causes shifts in relative momenta.

Fractional shift $ heta/\pi$ is a conserved quantum number.

Abstract

I give a non-technical account of fractional statistics in one dimension. In systems with periodic boundary conditions, the crossing of anyons is always uni-directional, and the fractional phase $\theta$ acquired by the anyons gives rise to fractional shifts in the spacings of the relative momenta, ${\Delta p =2\pi\hbar/L\, (|\theta|/\pi+n)}$. The fractional shift $\theta/\pi$ is a good quantum number of interacting anyons, even though the single particle momenta, and hence the non-negative integers $n$, are generally not.