Detailed Study of an Ultra-Small Pauli Crystal

Particular geometric arrangements, called Pauli crystals, have been conjectured to exist in a two-dimensional system of free fermions under harmonic confinement. The fermions are neutral fermionic atoms with frozen spins. The question whether such crystalline structures do really exist is open for discussion. We adopted an effective statistical interaction potential approach to study such structures in the past, but without conducting a detailed analysis of their stability. In this work, we point out the fact that the smallest Pauli crystal that contains three fermions can be studied exactly within the framework of our approach. This means that we can predict the structure, size and geometry at any temperature where the Pauli crystal, presumably, emerges. Our results for three fermions appear to be in good quantitative agreement with the reported values of Pauli crystals seen in single-shot imaging data at a specific temperature. However, it is remarked that the size of the crystal is temperature-dependent in our approach. This feature seems to have gone unobserved and/or unnoticed in earlier work.