# Magic Numbers and Mixing Degree in Many-Fermion Systems

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## Abstract

**:**

## 1. Introduction

#### Present Goal

## 2. Preliminaries

#### 2.1. Quantum Mixing-Degree Quantifier

#### 2.2. Usefulness of Exactly Solvable Many-Body Systems

- Insight into quantum phenomena: Exactly solvable many-body systems often serve as simple and tractable models that exhibit essential quantum phenomena, such as quantum phase transitions, entanglement, and quantum correlations. They provide valuable intuition and understanding of fundamental quantum concepts.
- Testing quantum theories: Because these systems are analytically solvable, they are ideal for testing and validating theoretical methods and approximations used in more complicated systems. They allow researchers to check the accuracy and efficiency of numerical algorithms and analytical techniques.
- Educational tools: Exactly solvable many-body systems are commonly used as educational tools in teaching quantum mechanics and statistical physics. They provide students with concrete examples to illustrate abstract concepts and principles.
- Foundation for approximations: Many-body systems that are exactly solvable often serve as the foundation for developing approximate methods applicable to more complex systems. These methods include mean-field theory, perturbation theory, and variational approaches.
- Condensed matter physics: Exactly solvable models play a crucial role in understanding phase transitions and critical phenomena in condensed matter physics. They shed light on the emergence of collective behaviors in large systems.
- Quantum information theory: Solvable models are essential in quantum information theory, particularly in studies related to quantum computing, quantum error correction, and quantum communication protocols.
- Benchmarking numerical techniques: Exactly solvable models provide precise results that can be used as benchmarks to assess the accuracy and efficiency of numerical techniques, such as Monte Carlo simulations, tensor network methods, and a density-matrix renormalization group (DMRG).

#### 2.3. Using Very Low Temperature Statistical Mechanics Techniques to Approximate Ground-State Properties

**which we use in this work**, is an essential tool for studying ground-state properties in various physical systems, including condensed matter physics, quantum chemistry, and quantum information theory. They allow researchers to gain insights into the behavior of complex quantum systems and provide a foundation for understanding and engineering quantum materials and technologies. Concomitant references are given below.

#### 2.4. Magic Numbers in Many-Fermion Systems

#### 2.5. Expanding on Our Present Objectives

## 3. The AFP Model Structure

#### 3.1. Quasi-Spin Operators

#### 3.2. The AFP Model

## 4. Working within the Gibbs Ensemble Framework

#### A State’s $\rho $ Degree of Mixture ${C}_{f}$

## 5. Present Results for Our Main Quantifier ${S}_{2}$

#### 5.1. Results as a Function of the Particle Number

**all v**and we encounter a special N value ($={N}_{m}$) for which ${S}_{2}$, and the mixing degree, suddenly grows. Here, we borrow the described “magic number” ${N}_{m}\left(v\right)$ from nuclear physics such that the system experiences a noticeable amount of mixing. Magic numbers are rather typical features of fermion systems. We discover that as v diminishes, ${N}_{m}$ grows.

#### 5.2. Energetic Interpretation of the ${N}_{m}$

#### 5.3. Results as a Function of the Coupling Constant v

#### 5.4. Effects of the ${S}_{2}$ Peaks on Macroscopic Quantities

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Our Hamiltonian Matrix

## References

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**Figure 1.**We plot ${C}_{f}={S}_{2}$ vs. N for several v-values, with $\beta =20$. Purity prevails, with intriguing exceptions. $v$—colors are assigned in this way: $v=0.5$ (violet); $v=0.3$ (rose); $v=0.2$ (brown); $v=0.1$ (grey); $v=0.05$ (orange); $v=0.03$ (blue); $v=0.01$ (black); $v=0.001$ (green); $v=0$ (red).

**Figure 3.**We plot ${C}_{f}={S}_{2}$ vs. v for $\beta =20$. Colors are as follows: $N=2$ (red); $N=4$ (blue); $N=6$ (green); $N=8$ (black); $N=10$ (orange). See that we confront here magic v-regions (windows), whose size diminishes as N grows. Outside these windows, the mixing degree vanishes.

**Figure 4.**We find ${C}_{f}={S}_{2}$ (

**left**), <U> (

**center**), and Shannon’s S (

**right**) confront vs. v for $N=2,4,\cdots ,10$, with $\beta =20$. One sees that <U> displays slope changes at the v values associated with entropic peaks. Regarding the trait <U>, this fact shows the existence of critical values for the coupling constants at which the mean energy suffers a slope change. These critical values are found within the areas covered by the ${S}_{2}$ peaks.

**Table 1.**Values of the energy difference $A\left(N\right)={E}_{1}-{E}_{0}$ for the number-of-particles triplets associated with the peaks in Figure 1. The values at the center of the triplet exhibit quasi-degeneracy as likely being responsible for the magic number peculiarity. That is, the two energies ${E}_{1}$ and ${E}_{2}$ are much closer to each other for ${N}_{m}$ than for ${N}_{m-1}$ or ${N}_{m+1}$.

Color Line | v | ${\mathit{N}}_{\mathit{m}}$ | ${\mathit{A}}_{\mathit{m}-2}$ | ${\mathit{A}}_{\mathit{m}}$ | ${\mathit{A}}_{\mathit{m}+2}$ |
---|---|---|---|---|---|

Black | 0.01 | 58 | 0.1629 | 0.0129 | 0.2243 |

Blue | 0.03 | 22 | 0.1959 | 0.1123 | 0.5503 |

Orange | 0.05 | 14 | 0.2689 | 0.0755 | 0.6447 |

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**MDPI and ACS Style**

Monteoliva, D.; Plastino, A.; Plastino, A.R.
Magic Numbers and Mixing Degree in Many-Fermion Systems. *Entropy* **2023**, *25*, 1206.
https://doi.org/10.3390/e25081206

**AMA Style**

Monteoliva D, Plastino A, Plastino AR.
Magic Numbers and Mixing Degree in Many-Fermion Systems. *Entropy*. 2023; 25(8):1206.
https://doi.org/10.3390/e25081206

**Chicago/Turabian Style**

Monteoliva, D., A. Plastino, and A. R. Plastino.
2023. "Magic Numbers and Mixing Degree in Many-Fermion Systems" *Entropy* 25, no. 8: 1206.
https://doi.org/10.3390/e25081206