Foundation of One-Particle Reduced Density Matrix Functional Theory for Excited States

Abstract

In Phys. Rev. Lett. 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems.Here, we develop a solid foundation for this so-called w-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case w = (1,0, •••). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.

An effective solution to convex $1$-body $N$-representability

“…Convexity Ansatz. The present article draws its motivation from this practical consideration, and the notion of convexity plays a key role: Convexity is essential in the foundation of reduced density matrix functional theory for excited states [SP21] [LCLS21]. Herein, we provide an effective solution to a convex relaxation of the constrained 1-body N -representability problem on which reduced density matrix functional theory relies, see Figure 6.…”

“…)). By resorting to Gross-Oliviera-Kohn variational principle in Theorem 1.5, we have recently generalized RDMFT to excited states where w is used to fix the weights of carefully chosen low-lying excited states [SP21] [LCLS21] . Application of exact convex relaxation turns wensemble RDMFT into a practically feasible method, provided a compact halfspace-representation of D 1 N (w) is found (which is nothing else than the ambition of the present work).…”

An exact one-particle theory of bosonic excitations: From a generalized Hohenberg-Kohn theorem to convexified N-representability

“…Intriguingly, the bosonic exchange statistics leads to substantial differences compared to fermionic w-ensemble RDMFT: It is one of the key achievements of our work to solve the underlying one-body w-ensemble N -representability problem. This in turn reveals a hierarchy of bosonic exclusion principles, the bosonic counterpart of Pauli's famous principle and its generalizations to mixed states [38,39].…”

“…In this section, we propose an w-ensemble RDMFT for targeting excited states of bosonic quantum systems and provide a comprehensive foundation for it. To keep our work self-contained, we recall the required key concepts from its fermionic counterpart [38,39], while focusing on the emerging challenges and modifications due to the bosonic exchange statistics.…”

“…A corresponding extended RDMFT was introduced for fermions in Ref. [38,39], as a generalization of the well-known DFT for excited states (GOK-DFT) [40][41][42]. The bosonic counterpart of this so-called w-ensemble RDMFT is missing so far.…”