Refining and relating fundamentals of functional theory

TL;DR

This paper refines the fundamental concepts of one-particle reduced density matrix functional theory (1RDMFT), clarifies its scope, explores symmetry and $v$-representability issues, and analytically investigates the properties of universal functionals for Hubbard models.

Contribution

It introduces a concise framework for 1RDMFT, relates fundamental features, and analytically derives properties of universal functionals for systems with symmetries and pair interactions.

Findings

Existence of six equivalent universal functionals under time-reversal symmetry

Analytical solutions for $v$-representability problems depending on pair interactions

Universal divergence of the gradient of functionals at domain boundaries

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.