Exact collective occupancies of the Moshinsky model in two-dimensional geometry

TL;DR

This work tackles the problem of exactly characterizing correlations in a 2D isotropic Moshinsky (harmonium) model of N bosons confined in a harmonic trap with harmonic interparticle interactions. The authors derive an exact Schmidt decomposition of the ground-state 1-RDM in polar coordinates, showing that angular parts are eigenfunctions of the angular-momentum operator and obtaining closed-form collective occupancies $\eta_l$ for each angular momentum $l$. They further analyze boson fragmentation by introducing $\eta_l$, the participation measure $K_\eta$, and the total participation $K$, revealing that correlations draw uniformly from all significant $l$ components and that fragmentation grows with the interaction strength $\Lambda$, with notable asymptotic behavior $\eta_l \sim \beta(N) \Lambda^{-1/4}$ and condensation trends $\lambda_{00} \to 1$ for large $N$. These results provide analytical benchmarks for quadratic-interaction approximations and deepen understanding of angular-momentum–resolved correlations and fragmentation in higher-dimensional bosonic systems.

Abstract

In this paper, we investigate the ground state of $N$ bosonic atoms confined in a two-dimensional isotropic harmonic trap, where the atoms interact via a harmonic potential. We derive an exact diagonal representation of the first-order reduced density matrix in polar coordinates, in which the angular components of the natural orbitals are eigenstates of the angular momentum operator. Furthermore, we present an exact expression for the collective occupancy of the natural orbitals with angular momentum $l$, quantifying the fraction of particles carrying that angular momentum. The present study explores how the dependence of collective occupancy relies on angular momentum $l$ and the control parameters of the system. Building on these findings, we examine boson fragmentation into components with different $l$ and reveal a unique feature of the system: the natural orbitals contributing to the correlations are uniformly distributed across all significant $l$ components.

Figures (3)

• Figure 1: Graphs (a) and (b) show the behaviors of collective occupancies as functions of interaction strength $\Lambda$ for two different particle numbers $N=2$ and $N=500$, respectively. The dashed lines represent the approximation results (\ref{['as']}). The strength of the interaction $\Lambda$ is expressed in units of $m \omega^2$. The graphs (c) and (d) illustrate the corresponding results for the participation $K_{\eta}$, together with its approximation obtained from the expansion to infinity, $\Lambda \to \infty$: $K_{\eta} \approx 2\beta^{-1}(N)\Lambda^{1/4}$ (dashed lines).
• Figure 2: Graph (a) shows the participation $K_{\eta}$ obtained for different values of $\Lambda$ as a function of $N$. The plot (b) shows the corresponding behaviors of the three lowest collective occupancies for $\Lambda=10^2$.
• Figure 3: Graphs (a) and (b) illustrate participation ratios $K$ and $\kappa_{l}$, respectively, for particle numbers $N=2, 50$ and $N=500$ as functions of $\Lambda$. The strength of the interaction $\Lambda$ is expressed in units of $m \omega^2$.