Pinning of Fermionic Occupation Numbers

Abstract

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.

Spectral properties of reduced fermionic density operators and parity superselection rule

“…Eigenvalues of 1-RDM are called natural occupation numbers. In general case, natural occupation numbers cannot be arbitrary ones within [0, 1] (simple Pauli's exclusion principle) and must satisfy generalized Pauli constraints [32][33][34][35].…”

“…In quantum chemistry, a particle reduction is realized by integrating over irrelevant particles' coordinates [28][29][30], which results in the so-called p-particle reduced density matrix (see, e.g., [31]). Such an approach provides new inequalities for fermionic occupation numbers and leads to the generalized Pauli constraints [32][33][34][35]. As far as systems with a varying number of particles are concerned, the construction of p-particle reduced density matrices is non-trivial and we discuss it in this paper.…”

Conditions for Describing Triplet States in Reduced Density Matrix Functional Theory

“…The class of spin-compensated systems with time-reversal symmetry is a notable exception, since, in that case, the conditions for pure-state N -representability collapse to the ensemble conditions [32]. The necessary and sufficient conditions for pure-state N -representability, also called generalized Pauli constraints, have only recently been discussed and explicitly expressed for systems with a small number of particles and specific finite sizes of the Hilbert space [33][34][35][36][37][38][39]. Recently, it has been demonstrated that with enforcing only the ensemble conditions in a RDMFT calculation for open-shell systems, the pure-state conditions will be violated for many functionals of the 1RDM [40].…”

Pauli Principle, Reloaded

“…Because a d-dimensional 1-particle Hilbert space can be imbedded into any (larger) d 0 -dimensional one, one sees that the convex polytope P N;d is nothing but the intersection between P N;d 0 and the set of spectra with only d nonzero eigenvalues (see also Ref. [15]). Hence, any facet of P N;d arises from the intersection of some facet of P N;d 0 with the subspace of said spectra.…”

“…Geometrically, these constraints define a convex polytope P N;d & R d of possible spectra (for more details see the Supplemental Material [15]). In general, if a spectral inequality such as (1) or (4) is (approximately) saturated, we say that the corresponding spectrum is (quasi)pinned to its extremum.…”