Contactium: A strongly correlated model system

Abstract

At the limit of an infinite confinement strength $\omega$, the ground state of a system that comprises two fermions or bosons in a harmonic confinement interacting through the Fermi--Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this ``contactium'' reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) $\{{\psi }_{\mathfrak{n}}(\omega ;\vec{r})\}$ of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their nonascendingly ordered occupation numbers $\{{\nu }_{\mathfrak{n}}\}$ turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties such as the ${\mathfrak{n}}^{-4/3}$ asymptotic decay of ${\nu }_{\mathfrak{n}}$ at the $\mathfrak{n}\to \infty$ limit (which differs from that of ${\mathfrak{n}}^{-8/3}$ in the Coulombic systems), the independence of the confinement energy ${v_\mathfrak{n} = \langle ψ_\mathfrak{n}(ω;\vec r) | \frac{1}{2} % ω^2r^2 | ψ_\mathfrak{n}(ω;\vec r) \rangle}$ of $\mathfrak{n}$, and the ${\mathfrak{n}}^{-2/3}$ asymptotic decay of the respective contribution ${\nu }_{\mathfrak{n}}{t}_{\mathfrak{n}}$ to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical with those of the two-electron harmonium atom at the $\omega \to \infty$ limit, despite the entirely different interparticle interactions in these systems.

9 pages, 5 figures, 1 table

References in corpus (3)

• Two electrons on a hypersphere: a quasi-exactly solvable model
• Correlation energy of two electrons in a ball
• Local and global interpolations along the adiabatic connection of DFT: A study at different correlation regimes