# Pauli Crystals–Interplay of Symmetries

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pauli Crystals

## 3. Few Particles in One Dimension

## 4. Closed Shells

## 5. Open Energy Shells

## 6. Particles in a Square Potential Well

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The one-particle density function ${\rho}^{\left(1\right)}\left(\mathit{r}\right)$ (

**a**,

**c**,

**e**) and the corresponding Pauli crystal (the configuration density function $\mathcal{C}\left(\mathit{r}\right)$) (

**b**,

**d**,

**f**) for $N=3,6$ and 10 fermions confined in one dimensional harmonic trap. Black dots correspond to the most probable configuration of particles (the pattern). All positions are scaled with respect to the natural oscillator length unit $\sqrt{\hslash /m\mathrm{\Omega}}$.

**Figure 2.**The one-particle density function ${\rho}^{\left(1\right)}\left(\mathit{r}\right)$ (left) and the corresponding Pauli crystal (the configuration density function $\mathcal{C}\left(\mathit{r}\right)$) (right) for $N=6$ fermions confined in an isotropic harmonic trap. Black dots correspond to the most probable configuration of particles (the pattern). Two geometrical shells containing one and five fermions are clearly visible. The color in the 2D plot gives the configuration density function $\mathcal{C}\left(\mathit{r}\right)$. Please note that the structure of energy shells is different. All positions are scaled with respect to the natural oscillator length unit $\sqrt{\hslash /m\mathrm{\Omega}}$.

**Figure 3.**The Pauli crystal (left) and the Coulomb crystal (right) for $N=15$ particles confined in the isotropic harmonic trap. Colors and black dots correspond to the configuration density function $\mathcal{C}\left(\mathit{r}\right)$ and the most probable configuration of particles, respectively. All positions are scaled with respect to the natural oscillator length unit $\sqrt{\hslash /m\mathrm{\Omega}}$.

**Figure 4.**Comparison of three dimensional Pauli crystals (top panel) and Coulomb crystals (lower panel) for $N=4$, 10 and 20 particles. Notice essential differences between the structures for 10 and 20 particles.

**Figure 5.**The one-particle density function $\mathcal{C}\left(\mathit{r}\right)$ (top) and the corresponding configuration density function $\mathcal{C}\left(\mathit{r}\right)$ (the Pauli crystal) (bottom) for $N=5$ particles confined in an isotropic harmonic trap. Left and right panels correspond to the circular (7) and cartesian (8) states, respectively. See the main text for details. Black dots indicate the most probable configuration (the pattern). All positions are scaled with respect to the natural oscillator length unit $\sqrt{\hslash /m\mathrm{\Omega}}$.

**Figure 6.**One-particle density function (first and third row) and corresponding configuration density function $\mathcal{C}\left(\mathit{r}\right)$ exposing Pauli crystals (second and fourth row) for $N=3,\dots ,8$ particles in a square well trap. Black points denote the most probable configuration of the many-body distribution ${\rho}^{\left(N\right)}$. Notice that the maxima of one particle density not always coincide with the position of particles forming the Pauli crystal. All positions are scaled with respect to the size of the well a.

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**MDPI and ACS Style**

Gajda, M.; Mostowski, J.; Pylak, M.; Sowiński, T.; Załuska-Kotur, M.
Pauli Crystals–Interplay of Symmetries. *Symmetry* **2020**, *12*, 1886.
https://doi.org/10.3390/sym12111886

**AMA Style**

Gajda M, Mostowski J, Pylak M, Sowiński T, Załuska-Kotur M.
Pauli Crystals–Interplay of Symmetries. *Symmetry*. 2020; 12(11):1886.
https://doi.org/10.3390/sym12111886

**Chicago/Turabian Style**

Gajda, Mariusz, Jan Mostowski, Maciej Pylak, Tomasz Sowiński, and Magdalena Załuska-Kotur.
2020. "Pauli Crystals–Interplay of Symmetries" *Symmetry* 12, no. 11: 1886.
https://doi.org/10.3390/sym12111886