# Quasi-Magical Fermion Numbers and Thermal Many-Body Dynamics

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## Abstract

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## 1. Introduction

## 2. Statistical Notion of Order

## 3. SU2-Based Fermion Models

#### 3.1. Lipkin Model

#### 3.2. The AFP Model

#### 3.3. Hamiltonian Matrices

## 4. Statistical Mechanics Indicators

#### Order Quantifiers

## 5. Singular Values for the Fermion Numbers

## 6. Conclusions

- The larger the value of N, the more stable the system becomes, as indicated by the behavior of the Lipkin free energy.
- This is not so in the AGFO case, where there is an absolute free energy minimum at a specific “magic” number.
- The main discovery with respect to the AFP entropy is the singular N value around N∼60 (the same as above). It signals not only stability but a loss of information.
- As regards Lipkin’s entropy, we have rather a lot to say. For small N numbers, the entropy almost vanishes, indicating a system in a mixed state close to the ground state. Then, and rather suddenly (magic number), as N grows, the system leaves the state described above and passes to a much more mixed state, losing information. The new state, however, remains stable as N continues to increase.
- AFPs D. The degree of order is large, in general, as the system lies in a state close to the ground state, as we saw above. However, for the same quasi-magic number N as above, the system abandons that state, passing to a much more mixed state and losing “order” as a consequence. This situation reverts back to the original one as N keeps growing.
- Lipkin’s D. The situation is much more complicated here than it was for the AFP model. For very small N values, this system lies in a disordered state. For moderate N values, a large degree of order is attained. Then, at the quasi-magic N value, the degree of order diminishes and then remains constant as N keeps increasing.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Lipkin free energy F vs. N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. The larger N is, the more stable the system becomes, as a low free energy is a sign of stability [16].

**Figure 2.**AFP free energy F vs. N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. Notice the minimum at N∼50 indicates maximal stability, as discussed in the text. Note that for the AFP model, things are quite different to the Lipkin case. This seems to show that a simple change in the fermion–fermion interaction form has a great effect on the free energy.

**Figure 3.**AFP entropy S versus N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. The main discovery here is the singular N value around N∼60. It signals a significant loss of information only for a very specific particle number, which we might call a quasi-magic number.

**Figure 4.**Lipkin entropy S vs. N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. Note that the precision of the curves is not too high. For small values of N, the entropy almost vanishes, indicating a system in a mixed state close to the ground state. Then, and rather suddenly as N grows, the system leaves the state described above and passes to a much more mixed state, losing information. The new state, however, remains stable as N continues to increase.

**Figure 5.**AFP D vs. N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. Remember that D measures order. This degree of order is large, in general, as the system lies in a state close to the ground state, as we saw above. However, for a quasi-magic number N, the system abandons that state, passing to a much more mixed one and losing “order” as a consequence. This situation reverts back to the original one as N keeps growing.

**Figure 6.**Lipkin D vs. N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. The situation is much more complicated here than it was for the AFP model. For very small values of N, this system lies in a disordered state. For moderate values of N, a large degree of order is attained. Then, at a quasi-magic N value, the degree of order diminishes and then remains constant as N keeps increasing.

**Figure 7.**AFP $\nu $ versus N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. Notice that around $N=60$, the energy cost $\nu $ abruptly becomes strongly negative, after being almost null for smaller values of N.

**Figure 8.**Lipkin $\nu $ vs. N for $V=0.01$ and $\beta =20$. V is a dimensionless variable and T is measured in Kelvins. The strong positive peak near $N=110$ implies that work is done on the system to make the degree of order decrease.

**Figure 9.**AFP $\nu $ versus N for $V=0.01$ and $\beta =30$. V is a dimensionless variable and T is measured in Kelvins. Notice that around $N=60$, the energy cost $\nu $ abruptly becomes strongly negative, after being almost null for smaller values of N.

**Figure 10.**Lipkin $\nu $ vs. N for $V=0.01$ and $\beta =30$. V is a dimensionless variable and T is measured in Kelvins. The strong positive peak near $N=110$ implies that work is done on the system to make the degree of order decrease.

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**MDPI and ACS Style**

Plastino, A.; Monteoliva, D.; Plastino, A.R.
Quasi-Magical Fermion Numbers and Thermal Many-Body Dynamics. *Axioms* **2023**, *12*, 493.
https://doi.org/10.3390/axioms12050493

**AMA Style**

Plastino A, Monteoliva D, Plastino AR.
Quasi-Magical Fermion Numbers and Thermal Many-Body Dynamics. *Axioms*. 2023; 12(5):493.
https://doi.org/10.3390/axioms12050493

**Chicago/Turabian Style**

Plastino, Angelo, Diana Monteoliva, and Angel Ricardo Plastino.
2023. "Quasi-Magical Fermion Numbers and Thermal Many-Body Dynamics" *Axioms* 12, no. 5: 493.
https://doi.org/10.3390/axioms12050493