Influence of the fermionic exchange symmetry beyond Pauli's exclusion principle

Pauli's exclusion principle has a strong impact on the properties of most fermionic quantum systems. Remarkably, the fermionic exchange symmetry implies further constraints on the one-particle picture. By exploiting those generalized Pauli constraints, we derive a measure which quantifies the influence of the exchange symmetry beyond Pauli's exclusion principle. It is based on a geometric hierarchy induced by the exclusion principle constraints. We provide a proof of principle by applying our measure to a simple model. In that way, we conclusively confirm the physical relevance of the generalized Pauli constraints and show that the fermionic exchange symmetry can have an influence on the one-particle picture beyond Pauli's exclusion principle. Our findings provide a perspective on fermionic multipartite correlation since our measure allows one to distinguish between static and dynamic correlations.

Figure 1

Left: Schematic illustration of the polytope $\mathcal{P}$, which is a proper subset of the light-gray “Pauli simplex” $\mathrm{\Sigma }$. The Hartree-Fock point is shown as a red dot. Right: For some generalized Pauli constraints $(D_1$ but not $D_2)$, a small distance of $\vec{\lambda }$ to the boundary of the surrounding Pauli simplex enforces a small distance $D_j\left(\vec{\lambda }\right)$ of $\vec{\lambda }$ to the polytope facet $F_j$.

Figure 2

For the Borland-Dennis setting, $(N,d)=(3,6)$, the reduced polytope of possible spectra $({\lambda }_4,{\lambda }_5,{\lambda }_6)$ is shown [the other three natural occupation numbers follow from Eqs. (16)]. It is spanned by the vertices $v_{HF}=(0,0,0),v_1=(1/2,1/2,1/2),v_2=(1/2,1/4,1/4)$, and $v_3=(1/2,1/2,0)$. Generalized Pauli constraint (17) takes the form ${\lambda }_5+{\lambda }_6-{\lambda }_4\ge 0$ and the Pauli simplex $\mathrm{\Sigma }$ is spanned by the vertices $v_{HF},v_1,v_3$, and $w$.

Figure 3

Schematic illustration of how the $L^2$ weight of $|\mathrm{\Psi }\rangle$ is distributed over various configurations. Higher weight is indicated by darker colors. Left: We use only the information of quasipinning by Pauli constraints. Middle: Stronger implications follow if additional nontrivial quasipinning by generalized Pauli constraints is taken into account. Right: Even if none of the Pauli constraints is (approximately) saturated, strong quasipinning by generalized Pauli constraints leads to structural simplifications (see text).

Figure 4

Quasipinning results for a three-harmonium ground state in two dimensions with external trap frequencies ${\omega }_1,{\omega }_2$ and coupling strength $\kappa$. Minimal distance $D_{min}$ of $\vec{\lambda }$ to the polytope boundary is shown on the left and its nontriviality is quantified on the right (see text for details).