Diverging exchange force and form of the exact density matrix functional - Schilling, C; Schilling, R (American Physical Society, 2019)
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 F depends only on the natural occupation numbers n. The respective sets P^1_N and 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 E^1_N = P^1_N, described by linear constraints D^{(j)}(n)⩾0. For smaller systems, it follows as F[n]=\sum_{i,i'} V_{i,i'} \sqrt{D^{(i)}(n)D^{(i')}(n)}. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a non-analytical term involving the interaction V. Third, the gradient dF/dn is shown to diverge on the boundary ∂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 V can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.