Generalized Pauli constraints in small atoms

The natural occupation numbers of fermionic systems are subject to nontrivial constraints, which include and extend the original Pauli principle. A recent mathematical breakthrough has clarified their mathematical structure and has opened up the possibility of a systematic analysis. Early investigations have found evidence that these constraints are exactly saturated in several physically relevant systems, e.g., in a certain electronic state of the beryllium atom. It has been suggested that, in such cases, the constraints, rather than the details of the Hamiltonian, dictate the system's qualitative behavior. Here, we revisit this question with state-of-the-art numerical methods for small atoms. We find that the constraints are, in fact, not exactly saturated, but that they lie much closer to the surface defined by the constraints than the geometry of the problem would suggest. While the results seem incompatible with the statement that the generalized Pauli constraints drive the behavior of these systems, they suggest that the qualitatively correct wave-function expansions can in some systems already be obtained on the basis of a limited number of Slater determinants, which is in line with numerical evidence from quantum chemistry.

Figure 1

Results of the pinning analyses for the variational ground states of the Be atom for different symmetry subspaces: artificial spin triplet using only $s$ orbitals (left) and fully polarized, i.e., spin quintet (right). Increasing the dimension $d^{\prime }$ of the underlying truncated setting confirms the absence of pinning since the truncation error (red error bars) is smaller than the distance $D_{\min }^{\prime }$ to the polytope boundary.

Figure 2

Results of different truncated pinning analyses for the variational ground states of the Li atom for different symmetry subspaces: spin doublet, i.e., overall ground state (left) and fully polarized, i.e., spin quadruplet (right). Presence of pinning cannot be ruled out since $D_{\min }^{\prime }$ is smaller than the respective truncation errors (red error bars).