Generalized Pauli constraints in large systems: the Pauli principle dominates

TL;DR

This paper demonstrates that in large fermionic systems, the Pauli principle overwhelmingly governs the allowed states, with generalized Pauli constraints becoming significant mainly in small, low-dimensional systems.

Contribution

It proves that the volume of the allowed eigenvalue polytope approaches the Pauli principle prediction in large systems, highlighting the limited impact of generalized constraints in high dimensions.

Findings

Polytope volume approaches Pauli principle prediction as dimension increases.

Generalized Pauli constraints are most restrictive in small, low-dimensional systems.

Corrections from generalized constraints are negligible in large systems.

Abstract

Lately, there has been a renewed interest in fermionic 1-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove this polytope's volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the 1-body space grows, and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.