Natural Orbitals and Occupation Numbers for Harmonium: Fermions vs. Bosons

TL;DR

This paper analytically compares the natural orbitals and occupation numbers of harmonically interacting bosons and fermions in a one-dimensional trap, revealing similarities at weak coupling and notable differences at strong coupling.

Contribution

It provides an analytical calculation of the 1-particle reduced density operator for both bosonic and fermionic ground states, highlighting the behavior of natural orbitals and occupation numbers across interaction strengths.

Findings

Bosonic natural orbitals are Hermite functions with Boltzmann-distributed occupation numbers.

Fermionic natural orbitals closely resemble bosonic ones at moderate coupling.

Differences between fermions and bosons are significant mainly for the largest occupation numbers.

Abstract

For a quantum system of N identical, harmonically interacting particles in a one-dimensional harmonic trap we calculate for the bosonic and fermionic ground state the corresponding 1-particle reduced density operator $\rho_1$ analytically. In case of bosons $\rho_1$ is a Gibbs state for an effective harmonic oscillator. Hence the natural orbitals are Hermite functions and their occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions with not too large couplings the natural orbitals coincide up to just a very small error with the bosonic ones. In case of strong coupling this still holds qualitatively. Moreover, the decay of the decreasingly ordered fermionic natural occupation numbers is given by the bosonic one, but modified by an algebraic prefactor. Significant differences to bosons occur only for the largest occupation numbers. After all the "discontinuity" at the "Fermi level" decreases with increasing coupling strength but remains well pronounced even for strong interaction.