Contactium: A strongly correlated model system
- URL: http://arxiv.org/abs/2303.14982v1
- Date: Mon, 27 Mar 2023 08:27:33 GMT
- Title: Contactium: A strongly correlated model system
- Authors: Jerzy Cioslowski, Berthold-Georg Englert, Martin-Isbj\"orn Trappe, and
Jun Hao Hue
- Abstract summary: We study the one-particle description of a system that comprises two fermions or bosons in a confinement interacting through the Fermi--Huang pseudopotential.
A detailed analysis of the one-particle description reveals several peculiarities that are not encountered in conventional model systems.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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