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 $\mathit{n}$. 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 $D^{(j)}(\mathit{n})\ge 0$. For smaller systems, it follows as $\mathcal{F}[\mathit{n}]=\sum _{i,i^{\prime }}{\overline{V}}_{i,i^{\prime }}\sqrt{D^{(i)}(\mathit{n})D^{(i^{\prime })}(\mathit{n})}$. This generalizes to systems of arbitrary size by replacing each $D^{(i)}$ by a linear combination of $\{D^{(j)}(\mathit{n})\}$ and adding a nonanalytical term involving the interaction $\widehat{V}$. Third, the gradient $d\mathcal{F}/d\mathit{n}$ 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 a nonfixed particle number as well and $\widehat{V}$ can be any $p$-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.

Figure 1

Weak and strong coupling asymptotes [Eq. (11)] (dashed lines) and exact functional $\mathcal{F}$ (solid line).

Figure 2

Left: Exact result for the ground state energy $E_0(u)$ (blue solid line) from the exact functional. The weak and strong coupling result from the functionals [Eq. (11)] is shown by the blue dashed lines. The result from PNOF5 and PNOF7(-) is presented by orange and red dots, respectively. Right: Relative error $\mathrm{\Delta }E/E_0$ as a function of $u$.