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arXiv:1911.00471 · Journal of Mathematical Physics

Generalized Pauli constraints in large systems: The Pauli principle dominates

Robin Reuvers

Published Mar 1, 2021• 0 citations
arXivHTMLPDF

Abstract

Lately, there has been a renewed interest in fermionic one-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove that this polytope’s volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the one-body space grows and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue that the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.

DOI
10.1063/5.0031419

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Lately, there has been a renewed interest in fermionic one-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove that this polytope’s volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the one-body space grows and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue that the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.
Postulated by Wolfgang Pauli in 1925 to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers (ni ≤ 1), thereby allowing no more than one electron in each quantum state. This is nothing more than a necessary and sufficient condition for a 1-body reduced density matrix to be the contraction of an N-body density matrix, provided that the sum of such occupations equals the number of particles. The pure N-representability problem asks whether given natural occupation numbers can arise from an antisymmetric pure N-particle state. Recent progress on such a problem establishes a systematic approach to a set of stronger constraints on the natural occupation numbers, not implied by the exclusion principle. In fact, for a pure quantum system the natural occupations satisfy a set of linear inequalities, known as generalized Pauli constraints, of the form with . This generalization of the Pauli exclusion principle has been widely used in many-body physics to study the influence of the fermionic exchange symmetry, beyond the original Pauli principle. In this chapter we review recent applications of the generalized Pauli exclusion principle and discuss some of the open problems.
Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman’s ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.
We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost “pinned.”
We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned."
Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman's ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.
Analytical evidence for the physical relevance of generalized Pauli constraints (GPCs) has recently been provided in Schilling et al. [Phys. Rev. Lett. 110, 040404 (2013)]: Natural occupation numbers $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\equiv}({\ensuremath{\lambda}}_{i})$ of the ground state of a model system in the regime of weak couplings $\ensuremath{\kappa}$ of three spinless fermions in one spatial dimension were found extremely close, at a distance ${D}_{\mathrm{min}}\ensuremath{\sim}{\ensuremath{\kappa}}^{8}$ to the boundary of the allowed region. We provide a self-contained and complete study of this quasipinning phenomenon. In particular, we develop tools for its systematic exploration and quantification. We confirm that quasipinning in one dimension occurs also for larger particle numbers and extends to intermediate coupling strengths, but vanishes for very strong couplings. We further explore the nontriviality of our findings by comparing quasipinning by GPCs to potential quasipinning by the less restrictive Pauli exclusion principle constraints. This allows us to eventually confirm the significance of GPCs beyond Pauli's exclusion principle.