Pinning of fermionic occupation numbers: Higher spatial dimensions and spin

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.

Implications of pinned occupation numbers for natural orbital expansions: I. Generalizing the concept of active spaces

“…Yet, given this it is quite remarkable that the vector n of NONs has just a tiny distance to the polytope boundary given by the eighth power of the coupling strength, k µ D 8 . A succeeding comprehensive and conclusive study of harmonic trap systems [22,[28][29][30][31][32] has confirmed that such quasipinning represents a genuine physical effect whose origin is the universal conflict between energy minimization and fermionic exchange symmetry in systems of confined fermions [30]. The presence of such quasipinning (or even pinning if the system's chosen Hilbert space is quite small) has been verified also in smaller atoms and molecules [33][34][35][36][37][38][39][40][41][42][43][44][45] ).…”

“…Moreover, corresponding states Yñ | could take the specific form(30) only with respect to highly distinctive bases of natural orbitals. Indeed, for any Yñ | with of the form (30) there are infinitely many allowed orbital rotations in the = n n subspace (leaving r 1 invariant) and changing the form(30) to(24), i.e. leading to a superposition of six rather than three configurations.…”

“…For the Hubbard model with three sites and three electrons the ground state was shown to exhibit pinning [41]. In the self-consistent expansion (8) it takes the form (30). The corresponding natural orbitals are given by the following spin-momentum states sñ k | ( s = = k 0, 1, 2, ) ñ = ñ ñ = ñ ñ = ñ ñ = ñ ñ = ñ ñ = ñ 1 0 , 2 1 , 3 0 , 4 2 , 5 2 , 6 1 .…”

On the time evolution of fermionic occupation numbers

“…Recently, it has been suggested that the generalized Pauli constraints may facilitate the development of more accurate functionals within density-matrix functional theory [41][42][43] . Since quasipinning (say, D j (n) ≈ 0) is approximately observed for several ground states, the quasipinning "mechanism" has attracted some attention in quantum chemistry and quantum-information theory [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] .…”

Role of the generalized pauli constraints in the quantum chemistry of excited states

“…A pure N ‐electron quantum system is representable by a single N ‐electron quantum‐mechanical wave function. Recently, the generalized Pauli constraints have been systematically derived for arbitrary numbers of electrons and orbitals and applied to closed, time‐independent systems such as atoms and molecules as well exciton dynamics in photosynthetic light harvesting …”