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Influence of the exchange symmetry beyond the exclusion principle

Felix Tennie

Published Jan 1, 2016• 4 citations
Publisher

Abstract

Since typically the physics of many body quantum systems is solely described and determined by pairwise interactions, the concept of reduced particle information is fundamentally relevant, particularly for indistinguishable particles. Determining the compatibility of r-particle reduced density operators (r-RDOs) with some N-particle state is known as the N-representability problem. The Pauli exclusion principle provides a set of necessary and sufficient conditions for the N-representability of fermionic 1-RDOs. However, pure N-representability requires a set of stricter conditions which are known as Generalised Pauli constraints (GPCs). In this thesis we investigate the influence and significance of the GPCs beyond the well-explored and established relevance of the Pauli exclusion principle. In particular, we analytically study the (quasi-)pinning effect which corresponds to a saturation of the GPCs and imposes structural simplifications on the N-fermion state. So far, nothing is known about its origin and mechanisms. We will close this gap by analytically examining the dependence of quasipinning on the particle number, the dimensionality and the spin polarisation in a representative physical model system. Our results suggest the notion of a microscopic Pauli pressure which is a consequence of the competition between energy minimisation and antisymmetry, and drives the system towards the boundary of the allowed region. Furthermore, the relevance of quasipinning beyond the wellknown Pauli exclusion principle is evaluted by the Q-measure. The outcomes provide first direct evidence for the need to incorporate GPCs into the physical description. To explore the role of the fermionic exchange symmetry beyond the Pauli exclusion principle from a more general perspective, we consider systems of hardcore bosons. Indeed, they exhibit a (lattice-based) exclusion principle, although their exchange symmetry is bosonic. Occupation numbers on oneparticle states other than the lattice basis can exceed one. We prove that the maximal occupation number is given by λmax = N/d(d - N + 1) where N, d are the numbers of particles and lattice sites. In addition, we derive necessary conditions for the N-representability of hardcore boson 1-RDOs and investigate the one-particle entropy boundaries. Finally, we present one-dimensional physical systems which macroscopically saturate λmax and exhibit (fractional) Bose-Einstein condensation.

DOI
10.5287/ora-xox4jjkr1

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Since typically the physics of many body quantum systems is solely described and determined by pairwise interactions, the concept of reduced particle information is fundamentally relevant, particularly for indistinguishable particles. Determining the compatibility of r-particle reduced density operators (r-RDOs) with some N-particle state is known as the N-representability problem. The Pauli exclusion principle provides a set of necessary and sufficient conditions for the N-representability of fermionic 1-RDOs. However, pure N-representability requires a set of stricter conditions which are known as Generalised Pauli constraints (GPCs). In this thesis we investigate the influence and significance of the GPCs beyond the well-explored and established relevance of the Pauli exclusion principle. In particular, we analytically study the (quasi-)pinning effect which corresponds to a saturation of the GPCs and imposes structural simplifications on the N-fermion state. So far, nothing is known about its origin and mechanisms. We will close this gap by analytically examining the dependence of quasipinning on the particle number, the dimensionality and the spin polarisation in a representative physical model system. Our results suggest the notion of a microscopic Pauli pressure which is a consequence of the competition between energy minimisation and antisymmetry, and drives the system towards the boundary of the allowed region. Furthermore, the relevance of quasipinning beyond the wellknown Pauli exclusion principle is evaluted by the Q-measure. The outcomes provide first direct evidence for the need to incorporate GPCs into the physical description. To explore the role of the fermionic exchange symmetry beyond the Pauli exclusion principle from a more general perspective, we consider systems of hardcore bosons. Indeed, they exhibit a (lattice-based) exclusion principle, although their exchange symmetry is bosonic. Occupation numbers on oneparticle states other than the lattice basis can exceed one. We prove that the maximal occupation number is given by λmax = N/d(d - N + 1) where N, d are the numbers of particles and lattice sites. In addition, we derive necessary conditions for the N-representability of hardcore boson 1-RDOs and investigate the one-particle entropy boundaries. Finally, we present one-dimensional physical systems which macroscopically saturate λmax and exhibit (fractional) Bose-Einstein condensation.
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.
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.
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.
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.
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 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.