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 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.

arXiv:1210.5531Physical Review LettersJan 22, 2013104 citations

Quantum Marginal Problem and its Physical Relevance

Christian Schilling

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.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.

arXiv:1507.00299Jan 1, 201422 citations

Pinning of fermionic occupation numbers: General concepts and one spatial dimension

Felix Tennie, Daniel Ebler, Vlatko Vedral +1 more

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.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.

arXiv:1602.05198Physical Review AApr 29, 201624 citations

Quasipinning and its relevance for N -fermion quantum states

Christian Schilling

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.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.

arXiv:1409.0019Physical Review AFeb 9, 201544 citations

Pinning of fermionic occupation numbers: Higher spatial dimensions and spin

Felix Tennie, Vlatko Vedral, Christian Schilling

The role of the generalized Pauli constraints (GPCs) in higher spatial dimensions and by incorporating spin degrees of freedom is systematically explored for a system of interacting fermions confined by a harmonic trap. Physical relevance of the GPCs is confirmed by analytical means for the ground state in the regime of weak couplings by finding its vector of natural occupation numbers close to the boundary of the allowed region. Such quasipinning is found to become weaker in the intermediate- and strong-coupling regime. The study of crossovers between different spatial dimensions by detuning the harmonic trap frequencies suggests that quasipinning is essentially an effect for systems with reduced spatial dimensionality. In addition, we find that quasipinning becomes stronger by increasing the degree of spin polarization. Consequently, the number of states available around the Fermi level plays a key role for the occurrence of quasipinning. This suggests that quasipinning emerges from the conflict between energy minimization and fermionic exchange symmetry.The role of the generalized Pauli constraints (GPCs) in higher spatial dimensions and by incorporating spin degrees of freedom is systematically explored for a system of interacting fermions confined by a harmonic trap. Physical relevance of the GPCs is confirmed by analytical means for the ground state in the regime of weak couplings by finding its vector of natural occupation numbers close to the boundary of the allowed region. Such quasipinning is found to become weaker in the intermediate- and strong-coupling regime. The study of crossovers between different spatial dimensions by detuning the harmonic trap frequencies suggests that quasipinning is essentially an effect for systems with reduced spatial dimensionality. In addition, we find that quasipinning becomes stronger by increasing the degree of spin polarization. Consequently, the number of states available around the Fermi level plays a key role for the occurrence of quasipinning. This suggests that quasipinning emerges from the conflict between energy minimization and fermionic exchange symmetry.

arXiv:1606.05334Physical Review AJul 27, 201621 citations

Hubbard model: Pinning of occupation numbers and role of symmetries

Christian Schilling

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 $U$ 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 $N$ -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.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 $U$ 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 $N$-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.

arXiv:1506.08833Physical Review BOct 29, 201533 citations

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.

U

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

N

-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.