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 $\mathbf{n}$. The respective sets ${\mathcal{P}}_N^1$ and ${\mathcal{E}}_N^1$ 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 ${\mathcal{E}}_N^1\equiv {\mathcal{P}}_N^1$, described by linear constraints $D^{(j)}(\mathbf{n})\ge 0$. For smaller systems, it follows as $\mathcal{F}[\mathbf{n}]={\sum }_{\mathbf{i},{\mathbf{i}}^{\prime }}{\overline{\mathbf{V}}}_{\mathbf{i},{\mathbf{i}}^{\prime }}\sqrt{{\mathbf{D}}^{(\mathbf{i})}(\mathbf{n}){\mathbf{D}}^{({\mathbf{i}}^{\prime })}(\mathbf{n})}$. This generalizes to systems of arbitrary size by replacing each $D^{(i)}$ by a linear combination of $\{D^{(j)}(\mathbf{n})\}$ and adding a non-analytical term involving the interaction $\hat{V}$. Third, the gradient $\mathrm{d}\mathcal{F}/\mathrm{d}\mathbf{n}$ is shown to diverge on the boundary $\partial {\mathcal{E}}_N^1$, 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.

arXiv commentpublished version

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