Statistical phases and momentum spacings for one-dimensional anyons

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 angle $\theta$. 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 $\Delta p=2\pi \hslash /L\left(\left|\theta \right|/\pi +\text{non-negative}\text{integer}\right)$ for a system of length $L$. This condition is the analog of the quantization of relative angular momenta according to $l_{\text{z}}=\hslash \left(-\theta /\pi +2\times \text{integer}\right)$ for two-dimensional anyons.

Figure 1

(Color online) Fractional statistics in 2D and in 1D.