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.
Categories: physics.chem-ph cond-mat.quant-gas quant-ph