Diverging exchange force and form of the exact density matrix functional

Abstract: 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, it follows as $\mathcal{F}[\bf{n}]=\sum_{i,i'} \overline{V}_{i,i'} \sqrt{D^{(i)}(\bf{n})D^{(i')}(\bf{n})}$. This generalizes to systems of arbitrary size by replacing each $D^{(i)}$ by a linear combination of ${D^{(j)}(\bf{n})}$ and adding a non-analytical term involving the interaction $\hat{V}$. Third, the gradient $\mathrm{d}\mathcal{F}/\mathrm{d}\bf{n}$ is shown to diverge on the boundary $\partial\mathcal{E}^1_N$, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an "exchange force". All findings hold for systems with non-fixed particle number as well and $\hat{V}$ can be any $p$-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.