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

Abstract

We investigate the properties of natural orbitals and their occupation numbers of the ground state of two non-interacting anyons characterised by the fractional statistics parameter $α$ and confined in a harmonic trap. We work in the boson magnetic gauge where the anyons are modelled 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 $n^{-(4+2α)}$, where $n$ 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 wavefunction. We explain that the same results apply to the fermion magnetic gauge.

Figures (4)

• Figure 1: The leading-order cusp in the function $\tilde{G}_\beta(r_1,r_2)$ is reproduced by the first three terms of its asymptotic expansion around $r_1=r_2$. The plot shows the function $\tilde{G}_\beta(r_1,r_2)$ vs. the first three terms of its leading-order asymptotic expansion \ref{['eq:I_asymp']} for $\beta=1/5$, $r_1=0.5$.
• Figure 2: The NAs of the $l=0$ sector for $\alpha\in\left\{\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\right\}$ and $M=400$. The number $n$ represents the index of the values $\sigma_{n}$. The solid lines show the outcome of the fifth order polynomial regression for each value of $\alpha$.
• Figure 3: The fitted values of $\mathcal{D}(\alpha)$ from fifth order polynomial regression for $\alpha\in\left\{\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1\right\}$ and $M=400$. They are in perfect agreement with the formula \ref{['eq:NA_asymptotics']}.
• Figure 4: Top row: the $l=0$ natural orbitals of index 1 (a), and 6 (b) for various values of the fractional statistics parameter $\alpha$. Frame (c) shows a comparison of the numerical and asymptotic forms of the natural orbital $n=30$, for which $l=0$ and $\alpha = 1/5$. The natural orbitals are observed to cross the axis $n$ times, where $n$ is the principal quantum number of the orbital. Bottom row: a comparison of the numerically computed natural orbitals to their corresponding asymptotic formulae for large $n$, all with $l=0$ and $\alpha = 1/5$. For small $r$ the deviations negligible. Frames (d), (e) and (f) show the oscillatory component of the natural orbitals, with $n=10$ (d), $n=40$ (e) and $n=80$ (f). For small $n$, the deviations from the asymptotic formula diverge as $r$ increases from $0$. This divergence is less pronounced for larger $n$.