Solving one-body ensemble N-representability problems with spin

TL;DR

The paper addresses how to characterize the admissible orbital one-body reduced density matrices for N-electron systems with spin symmetry under a general mixed-state weight vector w. It develops a spin-adapted, convex-geometry framework showing that these admissible occupancies form a polytope Σ_{N,S}(w) inside [0,2]^d, defined by linear inequalities on natural orbital occupations, and independent of M and d, with a hierarchical structure in the number r of nonzero weights in w. Methodologically, it combines representation theory, Hasse diagrams, vertex enumeration, and normal-cone analysis to derive both vertex and facet descriptions, yielding explicit spin-dependent w-ensemble constraints for r up to 3 and singlet-specific forms. The results provide a foundational cornerstone for ensemble density functional theories (EDFT) and w-ensemble RDMFT, enabling rigorous domain definitions and guiding approximations for excited-state targeting and higher-order RDM contractions in spin-symmetric systems.

Abstract

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies $n_i$ of orbitals $\varphi_i$ according to $0 \leq n_i \leq 2$. In this work, we first refine the underlying one-body $N$-representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness $\boldsymbol w$ of the $N$-electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope $Σ_{N,S}(\boldsymbol w) \subset [0,2]^d$. These constraints are independent of $M$ and the number $d$ of orbitals, while their dependence on $N, S$ is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

Figures (8)

• Figure 1: Commutative diagram illustrating the relation among the two non-convex sets $\mathcal{E}^N_{S,M}(\boldsymbol{w}), \mathcal{L}^1_{N,S,M}(\boldsymbol{w})$ and their convex hulls $\hbox{$\mathcal{E}$}^N_{S,M}(\boldsymbol{w}),\hbox{$\mathcal{L}$}^1_{N,S,M}(\boldsymbol{w})$ via the map $\mu(\cdot)$.
• Figure 2: Illustration of the relations between the most relevant sets of (reduced) density matrices and spectra introduced in Sec. \ref{['sec:science']}. For the sake of clarity, we also present as an intermediate level the corresponding set $\hbox{$\mathcal{E}$}^1_{N,S, M}(\boldsymbol{w})$ of full 1RDMs $\gamma$. Accordingly, the map $\mu(\cdot)$ introduced in Eq. \ref{['eq:map']} maps an $N$-fermion density matrix $\Gamma\in\mathcal{E}^N_{S,M}$ to its orbital 1RDM $\gamma_l$. The solution to the relaxed orbital one-body $\boldsymbol{w}$-ensemble $N$-representability problem characterizes the spectral set $\Sigma_{N,S}^{\downarrow}(\boldsymbol{w})$.
• Figure 3: Illustration of the Hasse diagram of partially ordered configurations for $r\leq 3$ satisfying the stability conditions for $S,N,d$ with $N=7, S=3/2$. (see text for more explanations).
• Figure 4: Illustration of Hasse diagram of partially ordered configurations for a singlet and $N=6$.
• Figure 5: Hasse diagram of partially ordered configurations for $N=d=3$ and $S, M=1/2$.