# An investigation of pre-crystalline order, ruling out Pauli crystals and introducing Pauli anti-crystals

## TL;DR

This paper refutes the existence of Pauli crystals in fermionic systems, introduces the concept of Pauli anti-crystals, and proposes neighbor counting statistics as a more reliable method to detect crystalline order in multiparticle quantum states.

## Contribution

It demonstrates that Pauli crystals are artifacts, introduces the novel concept of Pauli anti-crystals, and develops neighbor counting statistics for better characterization of multiparticle correlations.

## Key findings

- Pauli crystals do not exist in non-interacting fermion systems.
- Anti-crystalline structures can form where fermions avoid lattice points.
- Neighbor counting statistics effectively identify emerging crystalline order.

## Abstract

Fluid states of matter can locally exhibit characteristics of the onset of crystalline order. Traditionally this has been theoretically investigated using multipoint correlation functions. However new measurement techniques now allow multiparticle configurations of cold atomic systems to be observed directly. This has led to a search for new techniques to characterize the configurations that are likely to be observed. One of these techniques is the configuration density (CD), which has been used to argue for the formation of "Pauli crystals" by non-interacting electrons in e.g. a harmonic trap. We show here that such Pauli crystals do not exist, but that other other interesting spatial structures can occur in the form of an "anti-Crystal", where the fermions preferentially avoid a lattice of positions surrounding any given fermion. Further, we show that configuration densities must be treated with great care as naive application can lead to the identification of crystalline structures which are artifacts of the method and of no physical significance. We analyze the failure of the CD and suggest methods that might be more suitable for characterizing multiparticle correlations which may signal the onset of crystalline order. In particular, we introduce neighbour counting statistics (NCS), which is the full counting statistics of the particle number in a neighborhood of a given particle. We test this on two dimensional systems with emerging triangular and square crystal structures.