Duality of reduced density matrices and their eigenvalues
For states of quantum systems of N particles with harmonic interactions we prove that each reduced density matrix ρ obeys a duality condition. This condition implies duality relations for the eigenvalues λk of ρ and relates a harmonic model with length scales ${{\ell }_{1}},{{\ell }_{2}},\ldots ,{{\ell }_{N}}$ with another one with inverse lengths $1/{{\ell }_{1}},1/{{\ell }_{2}},\ldots ,1/{{\ell }_{N}}$. Entanglement entropies and correlation functions inherit duality from ρ. Self-duality can only occur for noninteracting particles in an isotropic harmonic trap.
CeO2−x nanorods with intrinsic urease-like activity
The large-scale production and ecotoxicity of urea make its removal from wastewater a health and environmental challenge. Whereas the industrial removal of urea relies on hydrolysis at elevated temperatures and high pressure, nature solves the urea disposal problem with the enzyme urease under ambient conditions. We show that CeO2−x nanorods (NRs) act as the first and efficient green urease mimic that catalyzes the hydrolysis of urea under ambient conditions with an activity (kcat = 9.58 × 101 s−1) about one order of magnitude lower than that of the native jack bean urease. The surface properties of CeO2−x NRs were probed by varying the Ce4+/Ce3+ ratio through La doping. Although La substit…
Diverging exchange force and form of the exact density matrix functional
For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered: First, within each symmetry sector, the interaction functional $\mathcal{F}$ depends only on the natural occupation numbers $\bf{n}$. The respective sets $\mathcal{P}^1_N$ and $\mathcal{E}^1_N$ of pure and ensemble $N$-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope $\mathcal{E}^1_N \equiv \mathcal{P}^1_N $, described by linear constraints $D^{(j)}(\bf{n})\geq 0$. For smaller systems,…
Number-parity effect for confined fermions in one dimension
For $N$ spin-polarized fermions with harmonic pair interactions in a $1$-dimensional trap an odd-even effect is found. The spectrum of the $1$-particle reduced density matrix of the system's ground state differs qualitatively for $N$ odd and $N$ even. This effect does only occur for strong attractive and repulsive interactions. Since it does not exists for bosons, it must originate from the repulsive nature implied by the fermionic exchange statistics. In contrast to the spectrum, the $1$-particle density and correlation function for strong attractive interactions do not show any sensitivity on the number parity. This also suggests that reduced-density-matrix-functional theory has a more su…