Foundation of one-particle reduced density matrix functional theory for excited states

TL;DR

This paper establishes a rigorous theoretical foundation for the w-RDMFT, a functional theory for excited states based on the one-particle reduced density matrix, extending previous ground state approaches with convex analysis and exclusion principles.

Contribution

It develops a formal framework for w-RDMFT, including variational principles, convex relaxation, and a hierarchy of exclusion constraints, advancing the theoretical understanding of excited state density matrix functionals.

Findings

Derived a universal functional for ensemble states with fixed weights.

Implemented an exact convex relaxation for the functional.

Outlined a hierarchy of generalized exclusion principle constraints.

Abstract

In [Phys. Rev. Lett. 127, 023001 (2021)] a reduced density matrix functional theory (RDMFT) has been proposed for calculating energies of selected eigenstates of interacting many-fermion systems. Here, we develop a solid foundation for this so-called $\boldsymbol{w}$-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 $\boldsymbol{w}$ 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 $\boldsymbol{w}=(1,0,\ldots)$. 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.