# Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory

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## Abstract

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## 1. Introduction

## 2. Hysteresis Phenomenon of the Unstable Critical Point in the Broken Symmetry Phase

## 3. Solitons and Tachyons in SSB

## 4. Where Are the Tachyons in the SSB Phase?

## 5. Discussion about Phenomena Within the Zone $\Delta T$

## 6. Discussion for Tachyons and Solitons at the Critical Point of QCD and Beyond

## 7. Conclusions—Further Investigations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Free energy $U\left(\phi \right)$ for the symmetric phase (green curve) and the SSB (blue curve) (also see text). The minima in blue curve are marked at ${\phi}^{*}=\pm \sqrt{\frac{{r}_{0}}{{u}_{0}}}$, for ${r}_{0}={u}_{0}=1$.

**Figure 2.**Invariant density diagrams for the 3D Ising numerical experiment (${20}^{3}$ lattice, ${N}_{iter}=200,000$ sweeps) for 4 different temperatures: (

**a**) the critical point at ${T}_{c}=4.545$. (

**b**) The two lobes at $T=4.48$ indicate the two stable fixed points. The separation of the lobes is not yet complete. This indicates the presence of the unstable critical point. (

**c**) At ${T}_{SSB}=4.44$, the SSB has been completed. This is the last presence of the unstable critical point. (

**d**) Below the SSB there is only one lobe; here, an example for $\mathrm{T}=4.42<{T}_{SSB}$ is shown. The unstable critical point does not exist anymore. The existence of this evolution has first been reported in [14], referring to the preparation of strong earthquake events, as well as in the random telegraph noise of nanoscale UTBB FD-SOI MOSFETs [15].

**Figure 3.**The finite size effect of the unstable critical point is present in the broken symmetry phase. The hysteresis zone $\Delta T$ values for the lattice lengths $L=8,10,13,16,20,23,26,30,34$ are presented. A power–law relationship between $\Delta T$ and $L$ of the form $\Delta T~{L}^{-1.5}$ exists. This means that in small distances the hysteresis phenomenon could be important.

**Figure 4.**The dot blue curve is produced if the SSB free energy ${U}_{SSB}\left(\phi \right)$ (blue curve in Figure 1) shifted by $U=\frac{{r}_{0}{}^{2}}{4{u}_{0}}$ (or $U=\frac{{m}^{4}}{4\lambda}$ for field theories); the green curve represents free energy $U\left(\phi \right)$ for the symmetric phase (as in Figure 1).

**Figure 6.**(

**a**) The kink soliton at ${\mathrm{T}}_{\mathrm{SSB}}$ in the magnetization of 3D Ising model localized in time. (

**b**) Within the hysteresis zone, $\left({\mathrm{T}}_{\mathrm{SSB}}<\mathrm{T}<{\mathrm{T}}_{\mathrm{c}}\right)$ the kink soliton recognizes two components which are expressed by the fluctuations within the new vacua, as in case (

**a**), and fluctuations around the critical point $M=0$. The red dashed horizontal line at $M=0$ is provided to guide the eye. The scales in y-axis as well as in x-axis are the same in both plots for comparison reasons (see text for interpretation of these fluctuations).

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**MDPI and ACS Style**

Contoyiannis, Y.; Hanias, M.P.; Papadopoulos, P.; Stavrinides, S.G.; Kampitakis, M.; Potirakis, S.M.; Balasis, G. Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory. *Symmetry* **2021**, *13*, 1358.
https://doi.org/10.3390/sym13081358

**AMA Style**

Contoyiannis Y, Hanias MP, Papadopoulos P, Stavrinides SG, Kampitakis M, Potirakis SM, Balasis G. Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory. *Symmetry*. 2021; 13(8):1358.
https://doi.org/10.3390/sym13081358

**Chicago/Turabian Style**

Contoyiannis, Yiannis, Michael P. Hanias, Pericles Papadopoulos, Stavros G. Stavrinides, Myron Kampitakis, Stelios M. Potirakis, and Georgios Balasis. 2021. "Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory" *Symmetry* 13, no. 8: 1358.
https://doi.org/10.3390/sym13081358