Contactium: A strongly correlated model system

Abstract

At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in 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) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} 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 n−4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n−8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n−2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.

Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins “Conjecture”

“… where Φ( r⃗ ) = ∫∫ 2 Γ( x 1 , x 2 ; x 1 , x 2 ) δ( r⃗ 1 – r⃗ ) δ( r⃗ 2 – r⃗ ) d x 1 d x 2 is the on-top two-electron density and the NOs corresponding to different spin components are counted separately ( Figure 1 ). Similar power laws and the n → ∞ asymptotics of φ n ( r⃗ 1 ) are obtained for non-Coulombic systems such as the contactium 46 and the anyons. 47 Generalizations of the NOs, 48 such as the energy natural orbitals 49 − 51 and the natural transition 52 / binatural orbitals 53 are amenable to an analogous treatment.…”

“…The availability of this equation has opened an avenue to facile derivation (which otherwise is quite complicated − ) of various asymptotic power laws, e.g. lim n → ∞ nobreak0em.25em⁡ n 8 / 3 .1em ν n = ( 2 1 / 2 3 π 5 / 4 ∫ [ Φ false( r⃗ false) ] 3 / 8 d r⃗ ) 8 / 3 where Φ( r⃗ ) = ∫∫ 2 Γ( x 1 , x 2 ; x 1 , x 2 ) δ( r⃗ 1 – r⃗ ) δ( r⃗ 2 – r⃗ ) d x 1 d x 2 is the on-top two-electron density and the NOs corresponding to different spin components are counted separately (Figure ). Similar power laws and the n → ∞ asymptotics of φ n ( r⃗ 1 ) are obtained for non-Coulombic systems such as the contactium and the anyons . Generalizations of the NOs, such as the energy natural orbitals − and the natural transition/ binatural orbitals are amenable to an analogous treatment.…”

Characteristic features of strong correlation: lessons from a 3-fermion one-dimensional harmonic trap

“…The smallest scale is often described by many-body quantum mechanics, and in fact in many applications the quantum laws of physics govern some key behaviour. Among many-body quantum phenomena strong correlation is particularly challenging and has been widely studied, both in-silico and experimentally [1][2][3][4][5][6]. Stretched molecular bonds involved in biochemical processes and transition-metal catalysts for energy storage devices are some prominent examples of strongly-correlated problems that are relevant to the industry.…”