# Harmonic N-particle systems: reduced density operators and their properties

### Wednesday, 03 Sep. 2014, 10:00 - 11:30

**Presenter:** Dr. Christian Schilling; Atomic & Laser Physics, University of Oxford

**Host:** Dr. James Whitfield

**Where:** Ernst-Mach-Hörsaal, 2nd Floor, Boltzmanngasse 5, 1090 Vienna

Solving analytically the N-particle Schrödinger equation for interacting particles is typically impossible. One exception is given by harmonic models which are also relevant from a physical viewpoint since they arise as an effective description of lattice systems. We present two results. First, we prove that for any eigenstate of a harmonic system each of its *M*-particle reduced density operators *ρ** _{M}* obeys a duality condition. This condition implies duality relations for the eigenvalues

*λ*

*of*

_{k}*ρ*

*and relates a harmonic model with length scales*

_{M}*l*with another one with inverse lengths 1

_{1},….., l_{N}*/l*,….,1

_{1}*/l*. Entanglement entropies and correlation functions inherit duality from

_{N}*ρ*

*. Second, for the specific case of*

_{M}*N*identical particles we explore the influence of the exchange statistics on the 1-particle properties obtained from

*ρ*

*. Although the (natural) occupation numbers for fermions and bosons differ significantly the fermionic and bosonic natural orbitals are very similar.*

_{1}