Anyons in a tight wave-guide and the Tonks-Girardeau gas

We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter \alpha ∈ ]0,2[ α∈]0,2[ . In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2\pi \alpha 2πα , generating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisotropic trapping potential. This freezes the motion in the direction of strong trapping, leading to 1D physics along the weak direction. The latter is governed to leading order by the Tonks-Girardeau model of impenetrable bosons, independently of \alpha α .