Superfluid band calculations
for the neutron star inner crust

Kenta Yoshimura
Tokyo Institute of Technology

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INTRODUCTION

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Nuclear Electron Correspondence
Density Functional Theory

Nuclear systems
Hartree-Fock Eq.

Hohenberg-Kohn thm:

Kohn-Sham Eq.

Skyrme energy functional:

Nuclear ``matter’’:Band theory:

Effective mass:
Our Interest?

・Application of the band theory to the nuclear DFT・Thorough understandings of nuclear properties in the neutron star crust
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History of Band calculation
N.Chamel, PRC85, 035801 (2012)
possible drastic change of effective mass (entrainment)

Self-consistency
Phys. Rev. C 100, 035804 (2019)
Phys. Rev. C 105, 045807 (2022)

Superfluidity
Phys. Rev. C 94, 065801 (2016)

・still w/o pairing ・only 1 dim. crystal
・effective mass less than bare mass(anti-entrainment?)

Phys. Rev. Research 4, 033141 (2022)
・not self-consistent・overestimation w/o self-consistency?

Our final goal :１. develop ①self-consistent ②superfluid ③band theory

for all crystalline structures realized in the neutron star inner crust
２. perform calculations and extract the effective mass of free neutrons
３. obtain the effective mass as a function of baryon densities,
utilized for actual simulations for astronomical phenomena
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THEORY2024/2/23

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Band Theory
HFB theory
HFB equation:

Phys. Rev. C 100, 035804 (2019)

Densities:
Superfluid Band Theory for one dimensional crystal
Bloch wave number:

Bloch transformation:
HFB Eq:

Points・One-dimensional dependence
・Self-consistent equation
・Millions of orbitals (typically ~2400000)
Parallel computing with supercomputers
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Effective Mass Calculation
Without band

Band structure

Free neutron density“bound” neutron

potential
Bound or free?
With band
Neutrons

TDDFT :EoM :External Field

Protons

Calc effective mass by
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Conduction number density

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RESULT

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Calc. Results and Effective Masses

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Calculation Result : (nB = 0.05 fm-3, β equilibrium)(Upper : number density
lower : pairing field)
・Highly n-rich (Yp ~ 0.03), centro-localized・Both n & p have superfluidity (sup.conductivity)
・Most neutrons are dripped (about 70%)
Effective mass :… density of energetically free neutrons… neutrons effectively conducting freely
・Effective masses always less than baremass.
・Superfluidity slightly reduces eff. masses.

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EXTENSION

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Finite Temperature Extensions
F.T. Calculation is needed for Proto Neutron Star

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P.N.S (~MeV)

What we wanna know :・How is the neutrino N.S. cooling?
・When nuclear pasta has been cooked?
・How many neutrons thermally superfluid?
F.T. Calculation
Density calculation :Free Energy :Specific Heat :ν-pasta scattering
Neutrino Opacity :Structure factors :2024/2/23

neutrino-cooling(to ~keV)Finite Temperature Extensions

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F.T. structure :・Density change to uniform distribution
・Pairing goes to zero (normal fluid phase)density・Specific heat shows 2 phase transitions

Scattering calculation :・Opacity shows peak structure at tens MeV・Fluid phase doesn’t change behaviour
Pairing gap

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・At 5MeV (uniform) there’s no peakExtension for mag-field systems
Why magnetic-field?
・Observational data supports the existence of magnetar
・Theoretically predictedsurface mag-field

How can we investigate?
・Electrons are Randau-Labi quantized→Drastic change of nuclear composition?
・Recently effect on nucleus been investigated

What we wanna know
・How structure changes?
・How affect on Equation of State?
・How pairing dynamics appear?
(spin triplet superfluid, LOFF phase, etc…)
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Extension for mag-field systems
Formulation
Additional magnetic-field term

HFB equation with spin d.o.f. :Electron energy calculation
Beta-equilibrium condition :Charge neutrality :Electron energy level :Electron density :2024/2/23

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Calculation Result
Remark :図中Bは全てB★
数密度
・磁場によって変化（一様に近づく？）
・でもあまり変わらないかもギャップ
・陽子は一瞬で壊れる（常伝導に）・中性子は割と耐える→強磁場では壊れる

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Calculation Result
磁場を変数にPairing Gap、及び偏極度Pqを見る。

ギャップ
・陽子は一瞬で壊れる→B*～6000前後で変化？（誤差かも）・B*～6000で中性子は急激に変化→すぐにゼロにはならない、
B*～7000くらいにかけてゼロに

偏極度
・陽子：B*～400くらいで偏極はじまる→その後はほぼ線型に上がる
・中性子：B*～5000前後まではゼロ、その直後一気に上がる→その後は線型に上昇していく
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Calculation Result
B*=3000の偏極度分布

B*=10000の場合

・スラブ部分では陽子と逆方向に偏極・外側ではそれを打ち消すように偏極・全体ととして磁化0が保たれている（spin-spin interactionの影響？）
・同じくらいの割合で偏極2024/2/23

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Summary
What we’ve done
・developed the ①self-consistent ②superfluid ③band theory

for nuclear pasta in the neutron star inner crust・performed calculations and successfully extracted neutron effective mass

Extensions
・For finite temperature systems,
computing structure changes and ν-pasta scattering processes.
・For finite-magnetic-field systems,
computing spin polarization and phase transitions

What we’ll do・Extensions for two- or three- dimensional crystalline phases,
completing the “table” of effective mass as a function of density.

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