Solving one-body ensemble N-representability problems with spin

Julia Liebert; Federico Castillo; Jean-Philippe Labbé; Tomasz Maciazek; Christian Schilling

doi:10.22331/q-2025-12-02-1921

Solving one-body ensemble N-representability problems with spin

Julia Liebert^1,2, Federico Castillo³, Jean-Philippe Labbé⁴, Tomasz Maciazek⁵, and Christian Schilling^1,2

¹Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany
²Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany
³Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
⁴École de Technologie Supérieure, 1111 rue Notre-Dame Ouest, Montréal (Qc) H3C 6M8, Canada
⁵School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom

Published:	2025-12-02, volume 9, page 1921
Editor:	Jordi Tura
Eprint:	arXiv:2412.01805v2
Doi:	https://doi.org/10.22331/q-2025-12-02-1921
Citation:	Quantum 9, 1921 (2025).

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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.

Popular summary

The Pauli exclusion principle is one of the cornerstones of modern physics. It states that no two identical fermions such as electrons can occupy at the same time the same one-particle quantum state. This simple rule governs the structure of atoms, explains the stability of matter, and shapes the behavior of materials. Yet real electrons also carry spin and they are often found in mixtures of different quantum states, for example due to temperature or interactions. These additional features impose further restrictions on how electrons can occupy available states, but these refined rules have remained unclear.

In this work we uncover and describe these hidden restrictions. We show that once the spin of the system and the degree of mixedness of a many-fermion quantum state are taken into account, the allowed patterns of electron occupation form a much smaller and more intricate geometric region than the familiar Pauli hypercube. Using ideas from group theory, convex geometry, and combinatorics, we provide a complete description of this region for any number of electrons and any total spin.

This geometric understanding has important consequences. It supplies the missing foundation for new methods that aim to predict electronic excited states using only reduced information instead of full wavefunctions. Such methods are promising for chemistry and materials science where many systems are too large for wavefunction approaches. Our results specify exactly which one-body information can occur in a given spin sector and therefore help guide the development of accurate and efficient computational tools, particularly in density (matrix) functional theory. In this way our refined exclusion principles extend Pauli’s original exclusion principle and reveal a deeper geometric structure in the quantum world of many electron systems.

► BibTeX data

@article{Liebert2025solvingonebody,
  doi = {10.22331/q-2025-12-02-1921},
  url = {https://doi.org/10.22331/q-2025-12-02-1921},
  title = {Solving one-body ensemble {N}-representability problems with spin},
  author = {Liebert, Julia and Castillo, Federico and Labb{\'{e}}, Jean-Philippe and Maciazek, Tomasz and Schilling, Christian},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1921},
  month = dec,
  year = {2025}
}

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[1] Julia Liebert, Yannick Lemke, Murat Altunbulak, Tomasz Maciazek, Christian Ochsenfeld, and Christian Schilling, "Toolbox of spin-adapted generalized Pauli constraints", Physical Review Research 7 2, 023247 (2025).

[2] Julia Liebert, Anna O. Schouten, Irma Avdic, Christian Schilling, and David A. Mazziotti, "Refining ensemble N-representability of one-body density matrices from partial information", New Journal of Physics 27 12, 124511 (2025).

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