Anyon trajectories and the systematics of the three-anyon spectrum

TL;DR

This paper introduces the concept of trajectories in anyon spectra, describing how energy levels depend continuously on kinetic angular momentum, and systematically analyzes three-anyon trajectories including their crossings.

Contribution

It develops a unified framework for understanding anyon spectra through trajectories and systematically characterizes three-anyon trajectories, including their crossing behavior.

Findings

Trajectories consist of two straight segments connected by a nonlinear part.

Trajectories cross at bosonic/fermionic points with true crossings.

Systematics of three-anyon trajectories are established.

Abstract

We develop the concept of trajectories in anyon spectra, i.e., the continuous dependence of energy levels on the kinetic angular momentum. It provides a more economical and unified description, since each trajectory contains an infinite number of points corresponding to the same statistics. For a system of non-interacting anyons in a harmonic potential, each trajectory consists of two infinite straight line segments, in general connected by a nonlinear piece. We give the systematics of the three-anyon trajectories. The trajectories in general cross each other at the bosonic/fermionic points. We use the (semi-empirical) rule that all such crossings are true crossings, i.e.\ the order of the trajectories with respect to energy is opposite to the left and to the right of a crossing.