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arXiv:1409.0019 · Physical Review A

Quasipinning and its relevance for N -fermion quantum states

Christian Schilling

Published Feb 9, 2015• 44 citations
arXivHTMLPDF

Abstract

Fermionic natural occupation numbers (NON) do not only obey Pauli's famous exclusion principle, but are even further restricted to a polytope by the generalized Pauli constraints, conditions which follow from the fermionic exchange statistics. Whenever given NON are pinned to the polytope's boundary, the corresponding $N$-fermion quantum state $|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}$ simplifies due to a selection rule. We show analytically and numerically for the most relevant cases that this rule is stable for NON close to the boundary, if the NON are nondegenerate. In case of degeneracy, a modified selection rule is conjectured and its validity is supported. As a consequence, the recently found effect of quasipinning is physically relevant in the sense that its occurrence allows to approximately reconstruct $|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}$, its entanglement properties, and correlations from one-particle information. Our finding also provides the basis for a generalized Hartree-Fock method by a variational ansatz determined by the selection rule.

DOI
10.1103/physreva.91.022105

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Fermionic natural occupation numbers (NON) do not only obey Pauli's famous exclusion principle, but are even further restricted to a polytope by the generalized Pauli constraints, conditions which follow from the fermionic exchange statistics. Whenever given NON are pinned to the polytope's boundary, the corresponding NNN-fermion quantum state ∣\ensuremathΨN\ensuremath⟩|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}∣\ensuremathΨN​\ensuremath⟩ simplifies due to a selection rule. We show analytically and numerically for the most relevant cases that this rule is stable for NON close to the boundary, if the NON are nondegenerate. In case of degeneracy, a modified selection rule is conjectured and its validity is supported. As a consequence, the recently found effect of quasipinning is physically relevant in the sense that its occurrence allows to approximately reconstruct ∣\ensuremathΨN\ensuremath⟩|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}∣\ensuremathΨN​\ensuremath⟩, its entanglement properties, and correlations from one-particle information. Our finding also provides the basis for a generalized Hartree-Fock method by a variational ansatz determined by the selection rule.
When the natural occupation numbers of a fermionic state saturate a generalized Pauli constraint, the state obeys a selection rule: only those configurations of natural orbitals that saturate the constraint themselves contribute. Pinning thus singles out an active space. The selection rule was proved for non-degenerate occupation numbers; for degenerate ones it was conjectured, and proved for one saturated constraint under an assumption checked only for the constraints known at the time. Here, it is proved for every generalized Pauli constraint, with no assumption, and also in the spin-adapted setting. The rule can only fail for the ordering constraints λj≥λj+1λ_j\geλ_{j+1}λj​≥λj+1​. Several saturated constraints are served by one basis whenever their common zero face contains a non-degenerate point. In practice, occupation numbers are quasipinned rather than pinned, and the rule survives with an explicit error bound.
The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.
The Pauli exclusion principle as constraint on fermionic occupation numbers is a consequence of the much deeper fermionic exchange statistics. Just recently, it was shown by Klyachko that this antisymmetry of fermionic wave functions leads to further restrictions on natural occupation numbers. These so-called generalized Pauli constraints (GPC) significantly strengthen Pauli's exclusion principle. Our first goal is to develop an understanding of the mathematical concepts behind Klyachko's work, in the context of quantum marginal problems. Afterwards, we explore the physical relevance of GPC and study concrete physical systems from that new viewpoint. In the first part of this thesis we review Klyachko's solution of the univariate quantum marginal problem. In particular we break his abstract derivation based on algebraic topology down to a more elementary level and reveal the geometrical picture behind it. The second part explores the possible physical relevance of GPC. We review the effect of pinning, i.e. the saturation of some GPC by given natural occupation numbers and explain its consequences. Although this effect would be quite spectacular we argue that pinning is unnatural. Instead, we conjecture the effect of quasipinning, defined by occupation numbers close to (but not exactly on) the boundary of the allowed region. In the third part we study concrete fermionic quantum systems from the new viewpoint of GPC. In particular, we compute the natural occupation numbers for the ground state of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are strongly quasipinned, even up to medium interaction strengths. We identify this as an effect of the lowest few energy eigenstates, which provides first insights into the mechanism behind quasipinning.
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.
Fermionic natural occupation numbers do not only obey Pauli's exclusion principle, but are even further restricted by so-called generalized Pauli constraints. Such restrictions are particularly relevant whenever they are saturated by given natural occupation numbers $\stackrel{P\vec}{\ensuremath{\lambda}}=({\ensuremath{\lambda}}_{i})$. For few-site Hubbard models we explore the occurrence of this pinning effect. By varying the on-site interaction UUU for the fermions we find sharp transitions from pinning of $\stackrel{P\vec}{\ensuremath{\lambda}}$ to the boundary of the allowed region to nonpinning. We analyze the origin of this phenomenon which turns out be either a crossing of natural occupation numbers ${\ensuremath{\lambda}}_{i}(U),{\ensuremath{\lambda}}_{i+1}(U)$ or a crossing of NNN-particle energies. Furthermore, we emphasize the relevance of symmetries for the occurrence of pinning. Based on recent progress in the field of ultracold atoms our findings suggest an experimental setup for the realization of the pinning effect.
The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states.It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle.Here, we provide the first analytic analysis of the physical relevance of these constraints.We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential.Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned).The result suggests that the physics behind the phenomenon is richer than previously appreciated.In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy.Our findings suggest a generalization of the Hartree-Fock approximation.