Refining and relating fundamentals of functional theory

Abstract

To advance the foundation of one-particle reduced density matrix functional theory (1RDMFT) we refine and relate some of its fundamental features and underlying concepts. We define by concise means the scope of a 1RDMFT, identify its possible natural variables and explain how symmetries could be exploited. In particular, for systems with time-reversal symmetry, we explain why there exist six equivalent universal functionals, prove concise relations among them and conclude that the important notion of v-representability is relative to the scope and choice of variable. All these fundamental concepts are then comprehensively discussed and illustrated for the Hubbard dimer and its generalization to arbitrary pair interactions W . For this, we derive by analytical means the pure and ensemble functionals with respect to both the real-and complex-valued Hilbert space. The comparison of various functionals allows us to solve the underlying v-representability problems analytically and the dependence of its solution on the pair interaction is demonstrated. Intriguingly, the gradient of each universal functional is found to always diverge repulsively on the boundary of the domain. In that sense, this key finding emphasizes the universal character of the fermionic exchange force, recently discovered and proven in the context of translationally-invariant one-band lattice models.

Deriving density-matrix functionals for excited states

“…In case F w is convex, every γ ∈ E 1 N (w ) 1 is w -ensemble v-representable, a statement that is well known in the context of ground state functional theory (see, e.g., Ref. [20,59]). Since the Legendre-Fenchel transform of F w is the energy E w up to minus signs, this implies that for a convex functional F w the biconjugate [60] F * * w ≡ conv(F w ) ≡ F w is equal to the functional F w itself.…”

“…Since F w is given on its domain of v-representable η by min p ′ F w ,p ′ and since the latter is typically not convex on E 1 N (w ) (recall ( 5)), it follows (see, e.g., [59] and Sec. 4.4) that some w -ensemble N -representable η are not w -ensemble v-representable.…”