Pinning of fermionic occupation numbers: Higher spatial dimensions and spin

Felix Tennie, Vlatko Vedral, and Christian Schilling

Phys. Rev. A 94, 012120 – Published 27 July 2016

Abstract

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.

Received 17 June 2016

DOI:https://doi.org/10.1103/PhysRevA.94.012120

Physics Subject Headings (PhySH)

Quantum formalism

General Physics

Authors & Affiliations

Felix Tennie¹, Vlatko Vedral^1,2, and Christian Schilling^1,*

¹Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
²Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543

^*christian.schilling@physics.ox.ac.uk

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Vol. 94, Iss. 1 — July 2016

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Images

Figure 1
Illustration of the mapping of $N$ -fermion quantum states (left) onto their vectors $\vec{λ}$ (right) of decreasingly ordered natural occupation numbers, giving rise to a polytope $P$ (dark grey). $P$ is a proper subset of the Pauli simplex $Σ$ (grey) defined by $1 \geq λ_{1} \geq \dots \geq λ_{d} \geq 0$ .
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Figure 2
Graphical illustration of the fermionic ground state $| Ψ^{(f)} 〉$ for the exemplary case of $n = 2$ spatial dimension and $N = 5$ fermions. In general, $| Ψ^{(f)} 〉$ is given by a single Slater determinant, obtained by filling the $N$ boxes with lowest energy respecting Pauli's exclusion principle (left), multiplied by a correlation term $e^{{\vec{X}}^{t} B \vec{X}}$ (right) for the center of mass $\vec{X} \equiv \frac{1}{N} (\vec{x_{1}} + \dots {\vec{x}}_{N})$ . Each box, labeled by $\vec{μ} = (μ_{1}, ..., μ_{n})$ describes a one-particle orbital given by a harmonic oscillator state in $n$ dimensions with corresponding frequencies ${\tilde{ω}}^{(α)}, α = 1, ..., n$ .
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Figure 3
Quasipinning results for the three-Harmonium ground state in two spatial dimensions with external trap frequencies $ω_{1}, ω_{2}$ and coupling strength $κ$ . Minimal distance $D_{min}$ of $\vec{λ}$ to polytope boundary is shown on the left and its nontriviality is quantified on the right. The solid line represents the boundary $κ_{crit} (χ)$ in between the two different ground-state configurations with the effectively one-dimensional one on the right. Dashed lines mark relevant crossing of NONs (see text for more details).
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Figure 4
Top: Minimal distance $D_{min}$ of $\vec{λ}$ to the polytope boundary as a function of the trap frequency detuning $χ \equiv ω_{2} / ω_{1}$ for fixed particle-particle interaction strengths $κ$ . Bottom: Relevance of the quasipinning by GPCs beyond Pauli's exclusion principle as measured by the $Q$ parameter.
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Figure 5
Quasipinning results for the four-Harmonium ground state in three dimensions with external trap frequencies $ω_{1}, ω_{2}, ω_{3}$ and coupling strength $κ$ . Minimal distance $D_{min}$ of $\vec{λ}$ to polytope boundary is shown in the upper diagram and its nontriviality $Q$ is quantified in the lower diagram (see text for details). The left and right solid curves indicate a change of the ground-state configuration.
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Figure 6
Quasipinning results for the non-fully-spin-polarized three-Harmonium ground state in two spatial dimensions with external trap frequencies $ω_{1}, ω_{2}$ , coupling strength $κ$ , and some fixed homogeneous magnetic field $B$ fulfilling $ℏ ω \sqrt{1 + κ} < c | B | < 2 ℏ ω \sqrt{1 + κ}$ . Minimal distance $D_{min}$ of $\vec{λ}$ to polytope boundary is shown in the left diagram and its nontriviality is quantified in the right diagram by the $Q$ parameter. The solid line marks the boundary between the effectively two- and one-dimensional ground-states configuration.
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