# Skyrme Crystals, Nuclear Matter and Compact Stars

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## Abstract

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## 1. Introduction

#### 1.1. The Skyrme Model

#### 1.2. Skyrme Matter

## 2. Skyrme Crystals

#### 2.1. Crystal Ansatz

#### 2.1.1. Face Centered Cubic Crystal of Skyrmions

- h is odd, k and l are even or h is even, k and l are odd,
- a, b, c are all odd or a, b, c are all even.

#### 2.1.2. Body Centered Cubic Crystal of Half-Skyrmions

- h, k are odd, l is even.
- a, b and c are even.
- ${\beta}_{abc}={\beta}_{bca}={\beta}_{cab}$.
- ${\alpha}_{hkl}=-{(-1)}^{\frac{h+k+l}{2}}{\alpha}_{khl}$.
- ${\beta}_{abc}=-{(-1)}^{\frac{a+b+c}{2}}{\beta}_{bac}$.

#### 2.1.3. Enhanced Face Centered Cubic Crystal of Skyrmions

- h is odd, k and l are even,
- a, b, c are all odd.

#### 2.2. Energy Curves

#### 2.3. Phase Transitions

#### 2.3.1. Low Density Phase Transition

#### 2.3.2. High Density Phase Transition

#### 2.3.3. Fluid-like Transition

#### 2.4. New Lattice Solutions

## 3. Isospin Quantization and Symmetry Energy

## 4. Kaon Condensate in Skyrmion Crystals

#### 4.1. The Kaon Condensate Effective Potential

#### 4.2. Quantum Corrections to the Effective Potential

## 5. Neutron Stars

#### 5.1. TOV System of Equations

#### 5.2. A Generalized Skyrme EOS

#### 5.3. Skyrme Crystal EOS

#### Scan of the Parameters

- Due to the perfect scaling property, the (adimensional) energy and isospin moment of inertia approximately satisfy the following expressions,$$\begin{array}{cc}\hfill E(L,{c}_{6},{c}_{0})& ={K}_{2}L+\frac{{K}_{4}}{L}+{c}_{6}\frac{{K}_{6}}{{L}^{3}}+{c}_{0}{K}_{0}{L}^{3},\hfill \end{array}$$$$\begin{array}{cc}\hfill \mathsf{\Lambda}(L,{c}_{6},{c}_{0})& ={\mathsf{\Lambda}}_{2}{L}^{3}+{\mathsf{\Lambda}}_{4}L+{c}_{6}\frac{{\mathsf{\Lambda}}_{6}}{L},\hfill \end{array}$$
- We fix the energy scale ${E}_{s}$ to a desired value in MeV. This is equivalent to fixing one of the three free parameters of the model, for instance, ${f}_{\pi}$.
- Then, we calculate ${L}_{0}$ by minimizing (102) and the values of ${E}_{0},{n}_{0},{S}_{0}$ and ${L}_{\mathrm{sym}}$ for different pairs of values $(e,{\lambda}^{2})$.
- When we find a set of parameters $({f}_{\pi},e,{\lambda}^{2})$ that fits the nuclear magnitudes within their respective errors of at most 15% then we calculate the corresponding EOS and solve the TOV system to obtain the mass–radius curve.
- Finally, we accept the sets of values that satisfy the constraints, ${M}_{\mathrm{max}}\ge 2{M}_{\odot}$ and ${R}_{1.4{M}_{\odot}}\le 12.5$ km. These constraints are motivated from pulsar measurements [78,85,86]. We find that there is more than one set of parameters, so there is a residual freedom in the choice of these values that satisfy the nuclear physics magnitudes at saturation and NS observables.

#### 5.4. Including Kaons

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

**This work is dedicated to the memory of our unforgettable friend and colleague Ricardo Vázquez.**

## Conflicts of Interest

## References

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**Figure 2.**Energy density contour plots of the face-centered cubic (FCC) unit cell of Skyrmions for a large value of $L=10$. Each plot shows different heights of the unit cell, which are $z=0$, $L/2$ and L because in this case also the energy density has periodicity in $2L$.

**Figure 4.**Energy density contour plots for the unit cell of the body-centered-cubic (BCC) crystal at the minimum of energy. In this case, the energy density has period L so we show the cuts at $z=0$, $z=L/4$ and $z=L/2$.

**Figure 5.**The tridimensional energy density plot of the BCC unit cell is shown. Again we represent the pion fields using the Runge coloring convention.

**Figure 6.**Energy density contour plots for the unit cell of the enhanced face-centered cubic crystal (FCC${}_{+}$) at the minimum of energy. It can be appreciated in the figure that this crystal can also be viewed as a simple cubic crystal of half-Skyrmions. The energy density has the same period as the BCC, so the cuts are the same.

**Figure 7.**The energy density in three dimensions for the FCC${}_{+}$ crystal. Again the Runge coloring convention is used as in the last two cases.

**Figure 8.**Energy curves of the different Skyrme crystals considering the different terms in the generalized Skyrme Lagrangian. We also show the BPS bound in each case to see how close it is to the minima of the crystals.

**Figure 9.**The mean value of the $\sigma $ field within the unit cell shows the second-order transition from the FCC to the FCC${}_{+}$ crystal when the pion-mass term is not included, and the asymptotic approach when it is included.

**Figure 10.**The comparison between the energies of the BCC and FCC crystals at the same baryon density shows that the first one becomes more favorable at some point, denoted by the red cross in the plots. The sextic term is crucial to have this transition at realistic densities. Both n and E are shown in Skyrme units.

**Figure 11.**Maxwell construction in the FCC-to-BCC phase transition. The red cross denotes again the transition point, but now the black crosses denote where the mixed phase starts. At low densities we are in the FCC phase (green line) until we reach the mixed phase (black line), which joins to the BCC phase (blue line).

**Figure 12.**REP for the ${\mathcal{L}}_{240}$ (blue) and ${\mathcal{L}}_{2460}$ (red) cases, for ${n}_{0}$ (continuous lines), $3{n}_{0}$ (dashed lines) and $7{n}_{0}$ (dashed-dotted lines). The degree of homogeneity almost does not change for ${\mathcal{L}}_{240}$, whereas the energy density becomes much more homogeneous with increasing baryon density for ${\mathcal{L}}_{2460}$.

**Figure 13.**Energy per baryon number for the different lattices that we consider with and without sextic term. These $\alpha $ and $B=32$ lattices are important in the low density regime where they have much lower energies than the standard crystal. However, already for densities which are slightly smaller than at the minimum of the energy all these lattices tend to the same FCC crystal.

**Figure 14.**Energy density contour of the $\alpha $ and $B=32$ lattices at different sizes of the unit cell. The shape of the isolated Skyrmion is visible at very large values of L, whereas the FCC${}_{+}$ crystal is recovered at high densities. The legend of the colors is as in Figure 6.

**Figure 15.**Fraction density ${\gamma}_{i}$ for each particle as a function of the baryon density for ${\lambda}^{2}=0$ (solid) and ${\lambda}^{2}=1.5$ (dashed). The corresponding DU threshold is also shown in black. This figure was originally published in [48].

**Figure 16.**Energy and baryon density of the standard Skyrme crystal, the BPS model, the Generalized model with ${p}_{PT}=50$ MeV/fm${}^{3}$ and the Hybrid model, with the same ${p}_{PT}$ and ${p}_{*}=2$ MeV/fm${}^{3}$. We also show some usually considered EOSs: APR4, WWF1, BCPM and we fill in orange and blue the regions of allowed values for ${p}_{PT}$ and ${p}_{*}$. This figure was originally published in [50].

**Figure 17.**Mass–Radius curves of the Hybrid model with the values ${p}_{PT}=\{25,40,50\}$, ${p}_{*}=\{0.5,1,2\}$ MeV/fm${}^{3}$. We compare our results with those of other EOSs and constraints obtained from different analyses. This figure was originally published in [50].

**Figure 19.**Energy and isospin moment of inertia of a Skyrmion crystal as obtained via the full numerical minimization (dots) and the perfect scaling fit (solid curves), for the models with and without sextic term.

**Figure 20.**Scan of the parameters $(e,{\lambda}^{2})$ for a fixed value of ${E}_{s}$. Each colored line corresponds to a fixed value of ${\lambda}^{2}$ and the black horizontal lines are the experimental values of ${E}_{0},{n}_{0}$ and ${S}_{0}$. The yellow dots are the optimal values and they lie in the black vertical line which is the corresponding value of e.

**Figure 22.**The EOS for the same 23 values shown before. In black, we plot the resulting EOS without isospin effects, whilst in red we consider $npe\mu $ matter. We find a good agreement between our EOS and the shaded regions obtained from the analysis in [37] at high densities. The purple region is an estimation for the range of the maximum central density inside NSs, and the dots represent the maximum central energy densities in our models.

**Figure 23.**Mass–Radius curves for the 23 representative sets of parameters considered. The colors of the lines represent the same as in Figure 22. The shaded regions correspond to GW (blue and orange) and pulsar (green) constraints.

**Figure 24.**Left plot: energy against the side length of the crystal, calculated with more points near the condensation values for both branches and their interpolations. Right plot: pressure against the energy density from which we conclude that there is a first-order phase transition. All magnitudes are shown in Skyrme units.

**Figure 25.**$E\left(L\right)$ curves for the two phases. The different slopes at the point of phase separation indicate a first-order phase transition. We also show the curves resulting from a Maxwell construction (MC) and a Gibbs construction (GC).

**Figure 26.**EOS for the three different cases that we have constructed. The jump in the MC due to the first-order transition and the different behavior of the GC at low densities are clearly visible. We also show the standard nuclear physics EOS of [75] (BCPM) and a hybrid EoS obtained by joining the BCPM EOS at low pressure with the GC EOS at high pressure. This figure was originally published in [49].

**Figure 27.**Particle fractions for the Gibbs Construction of the kaon condensation phase transition. This figure was originally published in [49].

**Figure 28.**Mass–Radius curves of NS with a kaon condensed core. The different sets of parameters that we consider are shown with different colors.

**Left**panel: Solid lines represent $npe\mu $ matter, dashed-dotted lines are obtained with an MC and the dashed with the GC.

**Right**panel: The effect of adding a standard nuclear physics crust to the Skyrme crystal EoS with kaon condensate obtained from the Gibbs construction (GC). This figure was originally published in [49].

Model | ${\mathit{L}}_{0}$ | ${\mathit{E}}_{0}/\mathit{B}$ | Minimum (%) |
---|---|---|---|

${\mathcal{L}}_{24}$ | 4.7 | 1.04 | 4 |

${\mathcal{L}}_{240}$ | 4.1 | 1.09 | 5 |

${\mathcal{L}}_{246}$ | 6.2 | 1.18 | 5 |

${\mathcal{L}}_{2460}$ | 5.3 | 1.30 | 7 |

Model | ${\mathit{L}}_{0}$ | ${\mathit{E}}_{0}/\mathit{B}$ | Minimum (%) |
---|---|---|---|

${\mathcal{L}}_{24}$ | 5.5 | 1.08 | 8 |

${\mathcal{L}}_{240}$ | 4.9 | 1.13 | 9 |

${\mathcal{L}}_{246}$ | 7.3 | 1.23 | 9 |

${\mathcal{L}}_{2460}$ | 6.4 | 1.34 | 10 |

**Table 3.**Fitting constants for the numerically obtained $E\left(L\right)$ curves for the FCC${}_{+}$ crystal.

Model | k | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{4}$ | ${\mathit{k}}_{6}$ | ${\mathit{k}}_{0}$ |
---|---|---|---|---|---|

${\mathcal{L}}_{24}$ | 0.047 | 0.105 | 2.344 | 0 | 0 |

${\mathcal{L}}_{240}$ | 0.022 | 0.109 | 2.384 | 0 | 0.008 |

${\mathcal{L}}_{246}$ | 0.006 | 0.106 | 2.750 | 0.905 | 0 |

${\mathcal{L}}_{2460}$ | 0.334 | 0.074 | 1.747 | 1.062 | 0.010 |

**Table 4.**Fitting constants for the numerically obtained $E\left(L\right)$ curves for the BCC crystal.

Model | k | ${\mathit{k}}_{2}$ | ${\mathit{k}}_{4}$ | ${\mathit{k}}_{6}$ | ${\mathit{k}}_{0}$ |
---|---|---|---|---|---|

${\mathcal{L}}_{24}$ | 0.017 | 0.096 | 2.988 | 0 | 0 |

${\mathcal{L}}_{240}$ | −0.022 | 0.101 | 3.061 | 0 | 0.004 |

${\mathcal{L}}_{246}$ | 0.040 | 0.093 | 3.150 | 1.710 | 0 |

${\mathcal{L}}_{2460}$ | 0.172 | 0.078 | 2.798 | 1.751 | 0.005 |

**Table 5.**Ratio between the transition density and the density at which the minimum of the energy is achieved for the FCC${}_{+}$ crystal.

Model | ${\mathit{n}}_{\mathbf{PT}}/{\mathit{n}}_{0}$ |
---|---|

${\mathcal{L}}_{246}$ | 3.7 |

${\mathcal{L}}_{2460}$ | 2.4 |

Label | ${\mathit{f}}_{\mathit{\pi}}$ | e | ${\mathit{\lambda}}^{2}$ | ${\mathit{E}}_{0}$ | ${\mathit{n}}_{0}$ | ${\mathit{S}}_{0}$ | ${\mathit{L}}_{\mathbf{sym}}$ | ${\mathit{n}}_{\mathbf{cond}}/{\mathit{n}}_{0}$ |
---|---|---|---|---|---|---|---|---|

set 1 | 133.71 | 5.72 | 5 | 920 | 0.165 | 23.5 | 29.1 | 2.3 |

set 2 | 138.11 | 6.34 | 5.78 | 915 | 0.175 | 24.5 | 28.3 | 2.2 |

set 3 | 120.96 | 5.64 | 2.68 | 783 | 0.175 | 28.7 | 38.7 | 1.6 |

set 4 | 139.26 | 5.61 | 2.74 | 912 | 0.22 | 28.6 | 38.9 | 1.6 |

n | 0 | 2 | 4 | 6 |
---|---|---|---|---|

${K}_{\mathrm{n}}$ | 0.034 | 0.466 | 9.617 | 4.329 |

${\mathsf{\Lambda}}_{\mathrm{n}}$ | – | 0.038 | 1.393 | 0.883 |

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## Share and Cite

**MDPI and ACS Style**

Adam, C.; Martín-Caro, A.G.; Huidobro, M.; Wereszczynski, A.
Skyrme Crystals, Nuclear Matter and Compact Stars. *Symmetry* **2023**, *15*, 899.
https://doi.org/10.3390/sym15040899

**AMA Style**

Adam C, Martín-Caro AG, Huidobro M, Wereszczynski A.
Skyrme Crystals, Nuclear Matter and Compact Stars. *Symmetry*. 2023; 15(4):899.
https://doi.org/10.3390/sym15040899

**Chicago/Turabian Style**

Adam, Christoph, Alberto García Martín-Caro, Miguel Huidobro, and Andrzej Wereszczynski.
2023. "Skyrme Crystals, Nuclear Matter and Compact Stars" *Symmetry* 15, no. 4: 899.
https://doi.org/10.3390/sym15040899