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arXiv · 2412.01805

Solving one-body ensemble N-representability problems with spin

Julia Liebert·Federico Castillo·Jean-Philippe Labbé·Tomasz Maciazek·Christian Schilling
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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 $\Sigma_{N,S}(\boldsymbol w) \subset [0,2]^d$. 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.

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Julia Liebert, Federico Castillo, Jean-Philippe Labbé, Tomasz Maciazek, Christian Schilling. 2024-12-02. Solving one-body ensemble N-representability problems with spin. https://doi.org/10.22331/q-2025-12-02-1921

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