# Superfluid Neutron Matter with a Twist

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Superfluid Neutron Matter: A Strongly Interacting Fermionic System

#### 2.1. The Variety of Approaches in Superfluid Neutron Matter

#### 2.2. The BCS and PBCS Theories for Neutron Matter

#### 2.2.1. Even-Particle-Number Superfluid

#### 2.2.2. Odd-Particle-Number Systems

#### 2.2.3. The Thermodynamic Limit

#### 2.2.4. The Solution of the BCS Gap Equations

## 3. Finite-Size Effects and Twisted Boundary Conditions

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NS | Neutron star |

NN | Neutron–neutron |

NM | Neutron matter |

BCS | Bardeen–Cooper–Schrieffer |

OES | Odd–even staggering |

PBCS | Projected BCS |

FBCS | Full BCS |

RG | Renormalization group |

QMC | Quantum Monte Carlo |

TL | Thermodynamic Limit |

PT | Pöschl–Teller |

CBF | Correlated basis functions |

FSE | Finite-size effect |

BC | Boundary condition |

TBC | Twisted boundary condition |

TABC | Twist-averaged boundary condition |

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**Figure 1.**The pairing gap at the thermodynamic limit (TL) divided by the Fermi energy as a function of the Fermi momentum ${k}_{\mathrm{F}}$. Additionally plotted are the pairing gaps for $\u2329N\u232a=66$ at ${k}_{\mathrm{F}}a=-10$ and $-5$.

**Figure 2.**The occupation probability distribution and the condensation amplitude for $\u2329N\u232a=66$ and ${k}_{\mathrm{F}}a=-10$.

**Figure 3.**The minimum of the quasi particle excitation energy and the odd–even staggering (OES) in the projected Bardeen–Cooper–Schrieffer (PBCS) theory under periodic boundary conditions (PBCs) for ${k}_{\mathrm{F}}a=-10$. The minimum of quasiparticle excitation energy under twist-averaged boundary conditions (TABCs) is shown in red.

**Figure 4.**The probability of finding a PBCS ground state with N particles projected out of a Bardeen–Cooper–Schrieffer (BCS) ground state with $\u2329N\u232a=66$ particles.

**Figure 5.**The twisted

**k**-space. Panel (

**a**) shows the

**k**-grid of a 2D system with N = 26 free neutrons under PBCs (squares) and twisted boundary conditions (TBCs) (circles) with

**θ**= 2π(1, 2)/7. The closed symbols represent the filled momentum states, and the closed curve shows the boundary of the filled

**k**-states of the infinite system. Panel (

**b**) shows the number of distinct

**k**-magnitudes generated by different twisted BCs.

**Figure 6.**The odd–even scattering (OES) for different twist angles $\mathbf{\theta}$. The angle $\mathbf{\theta}=2\pi (2,2,1)/7$ yields minimum finite-size effects (FSEs) in the OES across all N. The angle $\mathbf{\theta}=2\pi (0,1,0)/7$ yields minimum FSEs for $N=67$, a particle number routinely used in Quantum Monte Carlo (QMC) calculations for neutron matter (NM), as it has been demonstrated to yield minimum FSEs under PBCs.

**Figure 9.**The twist-averaged OES for ${k}_{\mathrm{F}}a=-10$. The twist-averaged minimum of the quasiparticle excitation energy from Figure 3 is also shown for comparison.

**Figure 10.**The variation $\sigma $ of the OES across the twist angles compared to the FSEs of the twist-averaged OES.

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Palkanoglou, G.; Gezerlis, A.
Superfluid Neutron Matter with a Twist. *Universe* **2021**, *7*, 24.
https://doi.org/10.3390/universe7020024

**AMA Style**

Palkanoglou G, Gezerlis A.
Superfluid Neutron Matter with a Twist. *Universe*. 2021; 7(2):24.
https://doi.org/10.3390/universe7020024

**Chicago/Turabian Style**

Palkanoglou, Georgios, and Alexandros Gezerlis.
2021. "Superfluid Neutron Matter with a Twist" *Universe* 7, no. 2: 24.
https://doi.org/10.3390/universe7020024