Statistical Phases and Momentum Spacings for One-Dimensional Anyons

TL;DR

This paper introduces a way to define fractional statistics for one-dimensional anyons by analyzing their momentum spacings, revealing a novel phase behavior consistent with topological considerations.

Contribution

It demonstrates that unidirectional crossings in 1D anyons allow for a consistent fractional phase, leading to quantized momentum spacings analogous to 2D anyon statistics.

Findings

Fractional momentum spacings depend on the statistical parameter theta.

Unidirectional crossings enable defining a fractional phase in 1D.

Momentum spacings are quantized as Delta p = 2 pi hbar/L (|theta|/pi + integer).

Abstract

Anyons and fractional statistics are by now well established in two-dimensional systems. In one dimension, fractional statistics has been established so far only through Haldane's fractional exclusion principle, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of one-dimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical parameter theta. The fractional statistics then manifests itself in fractional spacings of the single-particle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by Delta p = 2 pi hbar/L (|theta|/pi+non-negative integer) for a system of length L. This condition is the analogue of the quantisation of relative angular momenta according to l_z=hbar(-theta/pi+2integer) for two-dimensional anyons.