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 Schr\"odinger 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 ${v_\mathfrak{n} = \langle \psi_\mathfrak{n}(\omega;\vec r) | \frac{1}{2} % \omega^2r^2 | \psi_\mathfrak{n}(\omega;\vec r) \rangle}$ 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.