# BCS-BEC Crossover Effects and Pseudogap in Neutron Matter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. T-Matrix Formalism

#### 2.1. Vertex Function

#### 2.2. Numerical Solution for a Non-Separable Interaction

#### 2.3. Vertex Function with a Separable Interaction

#### 2.4. Self-Energy

#### 2.5. Occupation Numbers

## 3. Numerical Results

#### 3.1. Critical Temperature as a Function of the Chemical Potential

#### 3.2. Spectral Function and Pseudogap

#### 3.3. Occupation Numbers

#### 3.4. Correlated Density

#### 3.5. Density Dependence of the Critical Temperature

## 4. Open Questions

#### 4.1. Problem of the Subtraction

#### 4.2. Effect of the Quasiparticle Weight

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Feynman diagrams for (

**a**) the vertex function $\Gamma $ and (

**b**) the self-energy $\Sigma $ in ladder approximation.

**Figure 2.**BCS critical temperature ${T}_{c}$ as a function of the chemical potential $\mu $ for different cutoff values $\Lambda $. The red, green, purple, orange, and blue curves correspond to $\Lambda =1$, $1.5$, $1.8$, 2, and $2.5\phantom{\rule{0.277778em}{0ex}}{\mathrm{fm}}^{-1}$, respectively. The curves obtained for smaller cutoffs are only valid in the range where they agree with the $\Lambda =2$ and $2.5\phantom{\rule{0.277778em}{0ex}}{\mathrm{fm}}^{-1}$ results. The gray curve represents the critical temperature obtained with the separable potential (20).

**Figure 3.**Two-dimensional map of the spectral function $A(k,\omega )$ for $\mu =2\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ and $T=1.01\phantom{\rule{0.166667em}{0ex}}{T}_{c}\left(\mu \right)$ with the separable interaction. In figure (

**a**), we can clearly see the position of the peak located at $\omega ={k}^{2}/2-\mu $ and a double-peak structure for $k\lesssim {k}_{\mathrm{F}}$. Figure (

**b**) is a zoom on the region with the double peak.

**Figure 4.**Level densities for several values of the chemical potential $\mu =1\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ (blue line), $5\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ (green dash-dot line), $10\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ (red dashed line), and $20\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ (double-dashed purple line) slightly above the respective critical temperatures $T=1.01\phantom{\rule{0.166667em}{0ex}}{T}_{c}\left(\mu \right)$ which are $0.27,1.09,1.57$, and $1.53\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$.

**Figure 5.**Occupation numbers at $T=1.01\phantom{\rule{0.166667em}{0ex}}{T}_{c}\left(\mu \right)$ with (

**a**) $\mu =1\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ and (

**b**) $\mu =20\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$. The dotted black curve represents the free occupation numbers. The green dash-dot curve represents the NSR occupation numbers. The occupation numbers with HF subtraction (s-HF) and total subtraction (s-tot) are represented, respectively, by the red dashed curve and by the purple double dashed curve. Occupation numbers calculated with the full Dyson equation are represented by the blue line.

**Figure 6.**Correlated densities at $T=1.01\phantom{\rule{0.166667em}{0ex}}{T}_{c}\left(\mu \right)$ as a function of the chemical potential for four methods to calculate the correlations: (

**a**) Nozières and Schmitt-Rink (NSR), (

**b**) Hartree-Fock (HF) subtraction, (

**c**) total self-energy subtraction, and (

**d**) full Dyson equation. The calculations were done with ${V}_{\mathrm{low}\text{-}k}$ interactions corresponding to cutoffs $\Lambda =1$ (red), $1.5$ (green), $1.8$ (purple), 2 (orange), 3 (blue), and with the separable interaction (gray curves).

**Figure 7.**Critical temperature ${T}_{c}$ as a function of ${k}_{\mathrm{F}}$ calculated with the separable interaction with different approximations for the density: uncorrelated density (BCS, black solid line), NSR scheme (red dots), HF subtraction (green dashed-dotted curve), subtraction of the total self-energy (purple short dashes), full Dyson equation (orange long dashes).

**Figure 8.**Ratio ${T}_{c}/{E}_{\mathrm{F}}$ as a function of ${k}_{\mathrm{F}}$ for the separable interaction. The blue solid curve represents the free case (BCS), the red dotted curve the NSR case, the green dashed-dotted curve the HF subtraction, the purple short dashed curve the subtraction of the total self-energy, and the orange long dashed curve the density obtained with the full Dyson equation. Panel (

**a**) shows results for the physical $nn$ interaction, while panel (

**b**) shows results for the interaction that was readjusted to give an infinite $nn$ scattering length (unitary limit).

**Figure 9.**Real part of the on-shell self-energy $\Sigma (k,{\xi}_{k})$ as a function of k for $\mu =2\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$ and $T=1.01$${T}_{c}\left(\mu \right)=0.53\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}$, computed with the separable interaction (blue curve). The value of ${k}_{\mu}=\sqrt{2m\mu}/\hslash $ approximates the position of the Fermi surface. The red dotted line is not the result of a calculation but it is drawn by hand to show schematically how a more appropriate subtraction could look like.

**Figure 10.**Real part of $J(k=0,\phantom{\rule{0.166667em}{0ex}}\omega =0)$ for $\mu =20$ MeV as a function of the temperature T. The blue dash-dot curve represents the usual calculation, using $\overline{Q}$ according to Equation (5), while the the green curve displays ${J}^{\left(1\right)}$ which is obtained when the Fermi functions in the Pauli factor are replaced according to Equation (40). The red dashed curve is an extrapolation of ${J}^{\left(1\right)}$ down to ${T}_{c}^{\left(1\right)}$ under the assumption that it has the same form as J.

**Table 1.**Parameterizations of the separable interaction. First line: fit to the BCS critical temperature computed with ${V}_{\mathrm{low}\text{-}k}$ with cut-off $\Lambda =2\phantom{\rule{0.166667em}{0ex}}{\mathrm{fm}}^{-1}$ (${g}_{l=0}$ given here is related to $g=-856\phantom{\rule{0.277778em}{0ex}}\mathrm{MeV}\phantom{\rule{0.166667em}{0ex}}{\mathrm{fm}}^{3}$ in [30] by ${g}_{l=0}=gm/\left(4\pi {\hslash}^{2}\right)$). Second line: unitary limit ($a\to \infty $).

${\mathit{g}}_{\mathit{l}=0}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{fm}\right)$ | ${\mathit{q}}_{0}\phantom{\rule{0.277778em}{0ex}}\left({\mathbf{fm}}^{-1}\right)$ | |
---|---|---|

$nn$ interaction | $-1.644$ | $1.367$ |

unitary limit | $-1.834$ | $1.367$ |

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Durel, D.; Urban, M.
BCS-BEC Crossover Effects and Pseudogap in Neutron Matter. *Universe* **2020**, *6*, 208.
https://doi.org/10.3390/universe6110208

**AMA Style**

Durel D, Urban M.
BCS-BEC Crossover Effects and Pseudogap in Neutron Matter. *Universe*. 2020; 6(11):208.
https://doi.org/10.3390/universe6110208

**Chicago/Turabian Style**

Durel, David, and Michael Urban.
2020. "BCS-BEC Crossover Effects and Pseudogap in Neutron Matter" *Universe* 6, no. 11: 208.
https://doi.org/10.3390/universe6110208