Statistical interaction description of Pauli crystals in two-dimensional systems of harmonically confined fermions

It has been conjectured that the Pauli exclusion principle alone may be responsible for a particular geometric arrangement of confined systems of identical fermions even when there is no interaction between them. These geometric structures, called Pauli crystals, are predicted for a two-dimensional system of free fermions under harmonic confinement. It is assumed that the system consists of neutral fermionic atoms with their spins frozen (spin-polarized) in order to avoid any form of electromagnetic interaction. These crystalline patterns emerge as the most frequent configurations seen in a large collection of single-shot pictures of the system. In this work, we pursue the possibility of this outcome and consider a theoretical model that may capture both qualitatively and quantitatively key features of the above mentioned setup. Our approach treats a quantum system of non-interacting fermions as an effective classical system of particles that interact with an effective statistical interaction potential that mimics the quantum statistics. For this model, we consider two-dimensional few-body systems of harmonically confined particles that interact with a statistical potential and calculate analytically the minimum energy configuration for specific values of relevant parameters. The results for N=3 and 6 particles show that the minimum energy configuration corresponds to and is in good quantitative agreement with the reported values of Pauli crystals seen in single-shot imaging data obtained via the configuration density technique. Numerical results for larger systems of N=15 and 30 particles show that the crystalline configurations observed are not the same as the classical Wigner crystal structures that emerge should the confined charged particles interact with a Coulomb potential. An important question floated is whether such structures do really exist in a quantum system.