Structure of fermion nodes and nodal cells

TL;DR

This paper investigates the structure of fermion wave function nodes, proving that noninteracting spin-polarized fermions in harmonic wells have two nodal cells across various models, with interactions often reducing multiple cells to two.

Contribution

It extends the understanding of fermion node topology to interacting systems and various geometries, establishing conditions under which the minimal two nodal cells occur.

Findings

Spin-polarized noninteracting fermions in harmonic wells have two nodal cells.

Interactions tend to reduce multiple nodal cells to two.

Results apply to various models including fermions on a sphere and in periodic boxes.

Abstract

We study nodes of fermionic ground state wave functions. For 2D and higher we prove that spin-polarized, noninteracting fermions in a harmonic well have two nodal cells for arbitrary system size. The result extends to other noninteracting/mean-field models such as fermions on a sphere, in a periodic box or in Hartree-Fock atomic states. Spin-unpolarized noninteracting states have multiple nodal cells, however, interactions and many-body correlations generally relax the multiple cells to the minimal number of two. With some conditions, this is proved for interacting 2D and higher dimensions harmonic fermion systems of arbitrary size using the Bardeen-Cooper-Schrieffer variational wave function. Implications and extent of these results are briefly discussed.