Attractive statistical forces and Pauli crystal formation in trapped Fermi gases

Abstract

Exchange statistics endows identical particles with an effective "statistical potential", whose familiar exact form is two-body and purely repulsive for fermions. Here we construct an exact collective many-body form: the thermodynamics of $N$ trapped ideal fermions maps onto classical distinguishable particles governed by a single potential -- exactly for harmonic confinement at all temperatures, and to leading semiclassical order for arbitrary potentials. The associated force separates canonically into pairwise contributions, which for $N\geq 3$ can turn attractive, governed by a simple geometric criterion. Classical minimization reproduces observed few-body Pauli-crystal symmetries and agrees with the $N=55$ ground-state probability maximum at sub-percent shell accuracy. Heating drives discrete structural transitions accompanied by a crossover of the strongest force from attractive to repulsive. Both the potential and its forces are directly computable from existing single-shot imaging data, turning quantum exchange into measurable classical mechanics.