Natural orbitals and their occupation numbers for free anyons in the magnetic gauge

We investigate the properties of natural orbitals and their occupation numbers of the ground state of two noninteracting anyons characterized by the fractional statistics parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>α</a:mi></a:math> and confined in a harmonic trap. We work in the boson magnetic gauge where the anyons are modeled as composite bosons with magnetic flux quanta attached to their positions. We derive an asymptotic form of the weakly occupied natural orbitals, and show that their corresponding (ordered descendingly) occupation numbers decay according to the power law <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msup><b:mi>n</b:mi><b:mrow><b:mo>−</b:mo><b:mo>(</b:mo><b:mn>4</b:mn><b:mo>+</b:mo><b:mn>2</b:mn><b:mi>α</b:mi><b:mo>)</b:mo></b:mrow></b:msup></b:math>, where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>n</c:mi></c:math> is the index of the natural orbital. We find remarkable numerical agreement of the theory with the natural orbitals and their occupation numbers computed from the spectral decomposition of the system's wave function. We explain that the same results apply to the fermion magnetic gauge.