  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.
