YONUPA “Young Nuclear and Particle Physicist Group of Japan”
Summer School 2021: August 6-10, 2021 (via Zoom)
8/8 (Sun) 10:00-11:30 Sekizawa - Lecture 3

時間依存密度汎関数法で探る原子核ダイナミクス:
原子核反応から超流動現象，中性子星まで

Kazuyuki Sekizawa

Department of Physics, School of Science
Tokyo Institute of Technology

Lecture Plan

Theory (8/7 10:00-11:30)#1. An introduction to microscopic mean-field approaches and (TD)DFT

Nuclear reactions (8/7 13:00-14:30)#2. Recent advances in microscopic approaches for heavy-ion reactions
Superheavy element synthesis & deep-inelastic collisions

Neutron stars (8/8 10:00-11:30)#3. Neutron-star “glitch” and neutron superfluid
Dynamics of quantized vortices in the inner crust of neutron stars

Lecture Plan

Theory (8/7 10:00-11:30)#1. An introduction to microscopic mean-field approaches and (TD)DFT

Nuclear reactions (8/7 13:00-14:30)#2. Recent advances in microscopic approaches for heavy-ion reactions
Superheavy element synthesis & deep-inelastic collisions

Neutron stars (8/8 10:00-11:30)#3. Neutron-star “glitch” and neutron superfluid
Dynamics of quantized vortices in the inner crust of neutron stars

Voyage towards the limit of existenceNow we are sailing towards the edge of the nuclear landscape..

Let’s leave the planetof finite nuclei!

“Flat Earth” by iStock; “Star Wars Spaceship” by Valérian Pierret @Artstation

From nuclei to neutron stars“3d hyperspace background with warp tunnel effect Free Photo” @freepik

The Sun
・Radius: ~7×108m (~109 times bigger than Earth)・Mass: ~2×1030kg (~330 thousands times heavier than Earth)・Central temp.: ~10 million ℃・Surface temp.: ~5000 ℃

Stars shine due to nuclear fusion reactions
Picture: https://en.wikipedia.org/wiki/Sun

Energy source of the Sun: Nuclear fusion
Proton-proton (p-p) chain
1H

1H

1H

1H

e+

νe

e+

νe

2H

1H

1H

2H

3He

3He

1H

1H

4He

ProtonNeutronPositronEnergy source of the Sun: Nuclear fusion
Proton-proton (p-p) chain
1H

1H

1H

1H

e+

νe

e+

νe

2H

1H

1H

What happened:
Four protons became a helium nucleus!

2H

3He

3He

1H

Lighter than before!

1H

Before4He

ProtonNeutronPositron＞

AfterEnergy source of the Sun: Nuclear fusion
Proton-proton
Equivalence
between(p-p)
masschainand energy
1H

1H

1H

1H

e+

νe

e+

νe
E: Energy
m: Mass
2H
c: Speed of light

2H

1H

1H

What happened:
Four protons became a helium nucleus!

Albert Einstein (1879-1955)

3He

This reaction generates energy(exothermic)
1H

3He

Lighter than before!

1H

Before4He

ProtonNeutronPositron＞

After

Quantum tunneling allows forovercoming the Coulomb barrier!

The size is like Earth,
but with the solar massLife cycle of a light star

※The Sun will be a WD
after 5 billion years

Interstellar gas

White dwarf
Planetary nebula

Protostarwhen He is
exhaustedStar

Red giantHHe

p-p chain

Picture: https://www.astroarts.co.jp/products/stlnav7/spec/plane_list-j.shtml

Stability of Nuclei
EnergyMost stable
Lighter
after fission

Binding energy per nucleon (MeV)

56Fe

Heavy nuclei→ gain energy by nuclear fission
energy source of nuclear plants

Light nuclei→ gain energy by nuclear fusionenergy source of shining stars
EnergyLighter
after fusion

Mass number (= protons + neutrons)
Small

Large

The size is like Earth,
but with the solar massLife cycle of a massive star

※The Sun will be a WD
after 5 billion years

White dwarf

Interstellar gas

Planetary nebula

ProtostarStar
Red giantH

Red supergiantHe

p-p chain

Picture: https://www.astroarts.co.jp/products/stlnav7/spec/plane_list-j.shtml

The fate of a massive starcf. Lectures by
Dr. Thomas Chillery

Nuclear reactions:
1H → 4He → 12C → 16O → 20Ne → 24Mg → 28Si → ... → 56Fe
“Onion structure”

After forming the iron core…
→ no more fuel→ gravitational collapse→ supernova explosion

H

He
C, O
O, Ne, Mg
SiFe

The Crab Nebula
Remnant of the SN in 1054

Picture: https://en.wikipedia.org/wiki/Crab_NebulaLife cyclea starAnofultra-heavy,super-N-rich “nucleus”

The size is like Earth,
but with the solar mass

→ Paradise for nuclear physicists! :D※The Sun will be a WD
after 5 billion years

Interstellar gas

White dwarf
Planetary nebula

Protostar

Neutron star
SN remnant

Black holeStar

Red giant
H

Supernova

Red supergiantHe

p-p chain

Picture: https://www.astroarts.co.jp/products/stlnav7/spec/plane_list-j.shtml

Structure of a neutron star
Neutron star is a great playground for nuclear physicists
Ocean

Outer crust

Ocean Hot plasma~ 0.1 kmOuter crust

Nuclei (Fe), electronsρ < ρdrip

0.3-0.6 km

Inner crustρdrip~ 0.0014ρ0 ; Above ρdrip unbound neutrons exist outside nuclei
ρ0=2.8x1014 g/cm3=0.16 fm-3 ; Nuclear saturation density

Uniform nuclear matter:n, p, e-, μ-Inner crust

Nuclei, electrons,“dripped” neutrons

0.5-0.8 km

ρdrip ≤ ρ ≤ 0.6ρ0Hyperons?

Outer/inner core

Meson condensates?

9-12 km0.6ρ0 < ρ ≤ 3-5ρ0Quark matter?
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Structure of a neutron star
Neutron star is a great playground for nuclear physicists
Ocean

Outer crust

Ocean Hot plasma~ 0.1 kmOuter crust

Nuclei (Fe), electronsρ < ρdrip

0.3-0.6 km

Inner crustρdrip~ 0.0014ρ0 ; Above ρdrip unbound neutrons exist outside nuclei
ρ0=2.8x1014 g/cm3=0.16 fm-3 ; Nuclear saturation density

Uniform nuclear matter:n, p, e-, μ-Inner crust

Nuclei, electrons,“dripped” neutrons

0.5-0.8 km

ρdrip ≤ ρ ≤ 0.6ρ0Hyperons?

Outer/inner core

Meson condensates?

9-12 km0.6ρ0 < ρ ≤ 3-5ρ0Quark matter?
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Structure of a neutron star
Neutron star is a great playground for nuclear physicists
Ocean

Outer crust

Ocean Hot plasma~ 0.1 kmOuter crust

Nuclei (Fe), electronsρ < ρdrip

0.3-0.6 km

Inner crustρdrip~ 0.0014ρ0 ; Above ρdrip unbound neutrons exist outside nuclei“Nuclearρ =2.8x10
Pasta”14 g/cm
may3=0.16
exist!
fm-3 ; Nuclear saturation density0

Uniform nuclear matter:n, p, e-, μ-Inner crust

Nuclei, electrons,“dripped” neutrons

0.5-0.8 km

ρdrip ≤ ρ ≤ 0.6ρ0Hyperons?

Outer/inner core

Meson condensates?

9-12 km0.6ρ0 < ρ ≤ 3-5ρ0Quark matter?
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

What is Nuclear Pasta?

GnocchiLasagna

Spaghetti

M. E. Caplan and C. J. Horowitz, Rev. Mod. Phys. 89, 041002 (2017)

Equation of state (EoS) of nuclear matter➢ EoS characterizes the nuclear matter properties✓ From low-energy nuclear experiments, one may extract information in the vicinity of the saturation densityE.g.) ISGMR: U. Grag and G. Colò, PPNP101(2018)55; Electric dipole (E1) polarizability → skin thickness: A. Tamii et al., EPJA50(2014)28⇔ Esym. & L⇔ KA
Slope of the symmetry energy:

PNM: Pure Neutron Matter (n-only)
Symmetry energy:
SNM: Symmetric Nuclear Matter (N=Z)⇔ np > nn, pp & Pauli blocking

Saturation energy:

Density⇔ short-range attraction

Incompressibility:

Nuclear saturation density:⇔ short-range attraction & repulsive core

⇔ short-range attraction & repulsive corecf. Frontiers in Physics Vol. 21, Understanding the World of Supernova Explosion from Atomic Nuclei (原子核から読み解く超新星爆発の世界), K. Sumiyoshi (Kyoritsu printing Co., Ltd.)K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

From EoS to neutron stars➢ An EoS defines a Mass-Radius relation of neutron stars
 Equation of state, E(n)→ A relation between pressure and density:

 Tolman-Oppenheimer-Volkoff (TOV) equation:

where
Equilibrium condition:
Gravitational force (inward) + Pressure (outward) = 0

correction from General Relativity (GR): Integrated mass within radius r

Mass-radius relation

defines the NS radius

R (km)
Figure: P.B. Demorest et al., Nature 467, 1081 (2010)
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

How to solve the TOV equation - Ingredients
TOV eq.:Energy & Pressure

Skyrme EDF:

Energy density: Total energy par particle - often referred to as the equation of state (EOS): Pressure; and its derivative

Other definitions
Baryon (nucleon) number density:
Proton fraction:
Energy density for npeμ matter:
Mass density:

treated as relativistic
ideal Fermi gases

Chemical potential:
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

A recipe for solving the TOV equation
TOV eq.:*We know:

Central baryon density (initial condition at r = 0)

If

Determine Yp via the β-equilibrium condition (for npeμ matter):
Baryon density:
Proton fraction:: β equilibrium
Change Yp
Note that:

Integrate TOV eq.
by, e.g., Runge-Kutta

No: Charge neutrality

Yes: Mass density: Gravitational mass within radius r
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Particle fractions of npeμ matter

Chemical potentials vs. central density

β-equilibrium condition:

K. Sekizawa

Particle fractions vs. central density

Charge neutrality:

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Particle fractions of npeμ matter

Chemical potentials vs. central density

β-equilibrium condition:

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluid

Particle fractions vs. central density

Charge neutrality:

Sun., Aug. 8, 2021

Density and pressure as a function of radius✓ For various central densities, the TOV equation has been solved.

Baryon density vs. radius

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluid

Pressure vs. radiusSun., Aug. 8, 2021

Neutron star properties
Mass vs. central density*Something went wrong - it disagrees with published results.
Take it just as an example.

Mass-Radius relation

Radius vs. central density

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Neutron-star“glitch”

Picture: https://astronomy.com/magazine/ask-astro/2017/12/stellar-magnets

Pulsar - a rotating neutron star
✓ First discovery in August 1967 → “Little Green Man” LGM-1 → PSR B1919+21✓ Since then, more than 2650 pulsars have been observed✓ It gradually spins down due to the EM radiation

What is the glitch?

Typical example: the Vela pulsar➢ Irregularity has been observed from continuous monitoring of the pulsation period

Period (ms)
1970

89.25
89.24

Year
1975

1980

✓ Glitch: sudden increase of
the rotational frequency

Spin up!

89.23
89.22

Spin down

89.21
89.20

Time

Typical example: the Vela pulsar✓ Vela glitches roughly every 3 years(quasi-periodic)✓ Most of them are “large” (Δν/ν ~ 10-6),
but there are a few small glitches✓ 21 glitches have been observed
since its discovery in 1969

✓ Increase of spin down rate have also been
observed:~ 10-3-10-1*Δν: change of the rotation frequency due to a glitch
Vela: ν ~11.2 Hz (P ~89.3 ms)

Time between glitches (days)cf. Glitch Table and Glitch Catalogue

diagram for pulsars-10(*not necessarily glitchers)-12✓ Gradually spins down (due to the EM radiations-14✓ Period (P): milliseconds to seconds✓ Very stable “clock”, especially for
millisecond pulsars, i.e.

-16

)

-18

➢ Characteristic age:

➢ Surface dipole magnetic field strength:-20・ Binary pulsars

× High-energy pulsars
・ Radio pulsars
-3

-2

0

-1

(sec)

1

Figure taken from:
R.N. Manchester, J. Astrophys. Astr. 38, 42 (2017)

cf. Glitch Table and Glitch Catalogue

diagram for pulsars-10(*not necessarily glitchers)

Periodthan(P): 548
milliseconds
to seconds✓ Moreglitches havebeen
observed in more than 180 pulsars-12✓ Gradually spins down ()✓ Symbol size: glitch sizedue to the EM radiations(typically, log(Δν/ν) ~ 10-10-10-5)

-16

-14

Very stable“clock”,
especiallyfor✓ Youngpulsars(including
magnetars)
millisecond
exhibitlargerpulsars,
glitchesi.e.
than older ones-18

➢ Characteristic age:

➢ Surface dipole magnetic field strength:-20・ Binary pulsars

× High-energy pulsars
・ Radio pulsars
-3

-2

0

-1

(sec)

1

Figure taken from:
R.N. Manchester, Proc. IAU Symp. 337, 197 (2017)

What happened?

Something must happen inside the neutron star!
✓ “Starquake” model [G. Baym et al., Nature 224, 872 (1969)]

oblateless oblate
spin down

Preferable
shape✓ Sudden change of moment of inertia → Glitch?
It requires hundreds of years to have a next glitch..

✓ Vortex mediated glitch [P.W. Anderson and N. Itoh, Nature 256, 25 (1975)]➢ Dynamics of superfluid “quantized vortices” play a key role!

!?

Quantumvortices

Picture: N.P. Müller and G. Krstulovic, PRB102(2020)134513

What is a quantum vortex?
In superfluid, vortices are quantized!
Superfluid order parameter:

Superfluid velocity:

Vorticity:
superfluid is irrotational

Circulation:*Unless, there is no topological defectThis hole topologicallydistinguishes a mag from a cow

Pictures: https://en.wikipedia.org/wiki/Topology
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

What is a quantum vortex?
In superfluid, vortices are quantized!
Superfluid order parameter:

Superfluid velocity:

Vorticity:
superfluid is irrotational

Circulation:

Superfluid*Unless, there is no topological defectIf there is a defect:

Quantization of circulation*the phaseat the same point
must be equivalent!

Flow velocity of rotation
shall be quantized!

A hole at which
superfluidity is lost
Quantum vortex

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

What is a quantum vortex?
In superconductor, magnetic flux is quantized!
Magnetic flux:

Meissner effect: the London equation

If there is a defect:: density, mass, and charge ofa carrier (Cooper pair)

Quantization of magnetic flux (fluxtube, fluxoid, or fluxon)
Circulation:

Superfluid*Unless, there is no topological defectIf there is a defect:

Quantization of circulation*the phaseat the same point
must be equivalent!

Flow velocity of rotation
shall be quantized!

A hole at which
superfluidity is lost
Quantum vortex

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

In daily life, a vortex is continuous..

In superfluid, vortices are quantized!!

W. Ketterle, MIT Physics Annual. 2001

G.P. Bewley, D.P. Lathrop, and K.R. Sreenivasan, Nature 441, 588 (2006)

A movie from a talk by W. Guo (available from https://youtu.be/P2ckefSAN20) at
INT Program 19-1a “Quantum Turbulence: Cold Atoms, Heavy Ions, and Neutron Stars”
March 18 - April 19, 2019

Direct visualization of quantized vortices

Hydrogen particles were trapped in the vortex core, then worked as a tracer

Structure of a neutron star
Neutron star is a great playground for nuclear physicists
Ocean

Outer crust

Ocean Hot plasma~ 0.1 kmOuter crust

Nuclei (Fe), electronsρ < ρdrip

0.3-0.6 km

Inner crustρdrip~ 0.0014ρ0 ; Above ρdrip unbound neutrons exist outside nuclei
ρ0=2.8x1014 g/cm3=0.16 fm-3 ; Nuclear saturation density

Uniform nuclear matter:n, p, e-, μ-Inner crust

Nuclei, electrons,“dripped” neutrons

0.5-0.8 km

ρdrip ≤ ρ ≤ 0.6ρ0Hyperons?

Outer/inner core

Meson condensates?

9-12 km0.6ρ0 < ρ ≤ 3-5ρ0Quark matter?
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Structure of a neutron star
Neutrons (protons) are superfluid (superconducting) in neutron stars!
Ocean

Outer crust

Inner crust

2
1
0

0.1

Outer core1

A. Sedrakian and J.W. Clark, Eur. Phys. J. A 55, 167 (2019)

Inner crust: S-wave neutron superfluid
Inner core

Outer core: S-wave proton superconductor(+ P-wave neutron superfluid)

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Structure of the inner crust
A lattice of neutron-rich nuclei are immersed in a neutron superfluid
Quantum vortices can exist!
Pressureionization

Neutronization

Neutron drip

Nuclearpasta(Outside)(Inside)Fig.4 in N. Chamel and P. Haensel, Living Rev. Relativity 11, 10 (2008)
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Quantum vortices in a neutron star

In rotating superfluid, an array of quantum vortices is generated

 Observation in ultra-cold atomic gases

Vortex-number density
Low

Rotation speed

High

W. Ketterle, MIT Physics Annual. 2001
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Quantum vortices in a neutron star

In rotating superfluid, an array of quantum vortices is generated

 Observation in ultra-cold atomic gases

Vortex-number density
Low

Rotation speed

High

W. Ketterle, MIT Physics Annual. 2001
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Quantum vortices in a neutron star

There must be a huge number (~1018) of vortices inside a neutron star!!

Vortex-number density
Low

Rotation speed

High

W. Ketterle, MIT Physics Annual. 2001
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

How many vortices may be present inside a neutron star?
➢ For neutron vortices:cf. N. Chamel, J. Astrophys. Astr. 38, 43 (2017): number density: average distance➢ For flux tubes (proton vortices):: number density: average distance

For a typical neutron star (R=10 km, B=1012 G, P=10 ms), one finds:✓ Number of neutron vortices:✓ Number of flux tubes:

Why the core may not be important?

✓ Neutron vortices are magnetized in the core
Neutrons’ flow
Protons’ flow
*Protons are dragged byneutrons via nuclear force

Entrainment may increase neutron effective mass
→ inner crust may (or may not) be enough, see:- N. Chamel, PRC85(2012)035801- N. Anderson et al., PRL109(2012)241103- N. Chamel, PRL110(2013)011101- G. Watanabe and C.J. Pethick, PRL119(2017)062701- Yu Kashiwaba and T. Nakatsukasa, arXiv:1904.10712✓ Then, the neutron superfluid isstrongly coupled with the core due to:- magnetic field- scattering of electrons➢ If so, a “lag” can not be developed, and thus, the core
may not work as an angular momentum reservoir

The vortex mediated glitch: Naive picture

Rotation frequency

Superfluid neutrons(decoupled from the crust)cannot spin down if vortices
are “pinned” to the crust

Glitchmins to days

“Unpinning” occurs

Post glitch relaxation
days to years

Next glitch occurs
when sufficient lagis accumulated

Solid crust(strongly coupled to thecore by magnetic field)
spins downTime

To fully understand the glitches, we need to clarify:

Glitch dynamics

and, of course,
details of NS matter..

How do vortices move?

Pinning mechanism

Trigger mechanism

How are vortices pinned?

How are vortices unpinned?

We attacked this problem using
the state-of-the-art microscopic nuclear theory

Vortex-nucleus dynamics within TDSLDA

G. Wlazłowski, K.S., P. Magierski, A. Bulgac, and M.M. Forbes,
Phys. Rev. Lett. 117, 232701 (2016)

TDSLDA (Time-Dependent Superfluid Local Density Approximation)
TDSLDA: TDDFT with local treatment of pairing
Kohn-Sham scheme is extended for non-interacting quasiparticles➢ TDSLDA equations (formally equivalent to TDHFB or TD-BdG equations): number density: s.p. Hamiltonian: anomalous density: pairing field: current

A large number (104-106) of 3D coupled non-linear PDEs have to be solved!!
# of qp orbitals ~ # of grid points
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

TDSLDA (Time-Dependent Superfluid Local Density Approximation)
TDSLDA: TDDFT with local treatment of pairing
Kohn-Sham scheme is extended for non-interacting quasiparticles➢ TDSLDA equations (formally equivalent to TDHFB or TD-BdG equations): number density: s.p. Hamiltonian: anomalous density: pairing field: current

A large number (104-106) of 3D coupled non-linear PDEs have to be solved!!
# of qp orbitals ~ # of grid points
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

*The number indicates the rank according to the TOP500 list (June 28, 2021)
Piz Daint, CSCS, Switzerland (No. 15)

TITAN, ORNL, USA

TSUBAME3.0, Japan (No. 50)

Summit, ORNL, USA (No. 2)
GPU, 200 PFlops/s

Why GPUs?

CPU is like a very fast “sushi-production” machine

NOTE: the machine is an imaginary one
Pictures: Nikkan Kogyo Shimbun on Sep. 15, 2020 (https://newswitch.jp/p/23807); Kyouzan Sushi (https://www.kyouzan.jp/menu/)

CPU is like a very fast “sushi-production” machine
The speed is out of reach by a human power :(Pictures: PAKUTAS; Nikkan Kogyo Shimbun on Sep. 15, 2020 (https://newswitch.jp/p/23807); Kyouzan Sushi (https://www.kyouzan.jp/menu/)

What if we gather manpower?

Picture: 4.7” iPhone 6 Production to Start Next Week, July 18, 2014 (https://www.iphonefaq.org/archives/973661)

If a huge number of simple calculations are the bottleneck, and memory fits,a GPU may surpass a CPU!
✓ A single GPU can launch thousands of computing units in parallel✓ Memory (~16 GB/GPU) and data transfer cost (GPU-CPU) limit the applicability

CPU

GPU

Notice: TDSLDA equation is “local” in space✓ Each thread takes care of each spatial grid point.
✓ For spatial derivatives, we employ cuFFT which requires GPU-CPU data transfers.

GPU

CPU

Data transfer
for cuFFT➢ Orbitals are distributed totens to thousands of GPUs

*The number indicates the rank according to the TOP500 list (June 28, 2021)
Piz Daint, CSCS, Switzerland (No. 15)

TITAN, ORNL, USA

TSUBAME3.0, Japan (No. 50)

Summit, ORNL, USA (No. 2)
GPU, 200 PFlops/s

Present computing capabilities:

MPI Comm.

✓ Full 3D (w/o symmetry restrictions)
FFT✓ Volume as large as 1003 lattice points✓ Evolution up to 106 time steps (as long as 10-19 sec)

Applications of TDSLDATDSLDA has been successfully applied for both UFG and nuclear systems: s-wave scattering length: effective range

Unitary Fermi Gas (UFG)
A. Bulgac and S. Yoon, PRL102(2009)085302.
A. Bulgac et al., Science 332(2011)1288.
A. Bulgac et al., PRL108(2012)150401.
A. Bulgac et al., PRL112(2014)025301.
G. Wlazłowski et al., PRA91(2015)031602(R).
G. Wlazłowski et al., PRL120(2018)253002.
P. Magierski et al., PRA100(2019)033613.

Large-amplitude pairing field dynamics
Dynamics of quantum vortices
Quantum shock waves and domain walls
Dynamics of vortex rings
Dynamics of vortices and quantum turbulence
Solitonic cascades in spin-polarized UFG
Spin-polarized droplet in UFG

Nuclear systems
I. Stetcu et al., PRC84(2011)051309(R).
I. Stetcu et al., PRL114(2015)012701.
A. Bulgac et al., PRL116(2016)122504;
PRC100(2019)034615.
G. Wlazłowski et al., PRL117(2016)232701.
K. Sekizawa

Isovector giant dipole resonance (IVGDR)
Relativistic coulomb excitation
Induced fission of 240Pu
Vortex-nucleus interaction

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

A key to understand the glitches is:
Vortex pinning mechanism in the inner crust of neutron stars

Q. Is the vortex-nucleus interaction
Attractive?

“Nuclear pinning”

orRepulsive?

“Interstitial pinning”

What was the state-of-the-art?
Microscopic, static HFB calculations were performed assuming axial symmetry

Energy to create a vortex line
on a nuclear impurity

Energy to create a vortex line
in a uniform matterE.g.) 0.026 fm-3 (SLy4)
6.19 MeV

13058.04

12954.02

13714.88

13617.05

P. Avogadro, F. Barranco, R.A. Broglia, and E. Vigezzi, PRC75(2007)012805(R); NPA788(2007)130; NPA811(2008)378
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

What was the state-of-the-art?
Property of the pinning force was unclear

HFB (Avogadro et al.)Interstitial pinning

TF+LDA (Donati & Pizzochero)

Nuclear pinning

P. Avogadro, F. Barranco, R.A. Broglia, and E. Vigezzi, PRC75(2007)012805(R); NPA811(2008)378
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

What we investigated - Vortex-nucleus dynamics

Response of a spinning gyroscope when pushed

Response

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Method - TDSLDA
We performed 3D, dynamical simulations by TDDFT with superfluidity
Friedman & Pandharipande, NPA361(1981)502
Wiringa et al., PRC38(1988)1010

 TDSLDA equations (or TDHFB, TD-BdG)
PNM

SNM

 Energy density functional (EDF): Fayans EDF (FaNDF0) w/o LS

S.A. Fayans, JETP Lett. 68, 169 (1998)

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Method - TDSLDA
We performed 3D, dynamical simulations by TDDFT with superfluidity
Friedman & Pandharipande, NPA361(1981)502
Wiringa et al., PRC38(1988)1010

 TDSLDA equations (or TDHFB, TD-BdG)
PNM

SNM

 Energy density functional (EDF): Fayans EDF (FaNDF0) w/o LS

S.A. Fayans, JETP Lett. 68, 169 (1998)

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Method - TDSLDA
We performed 3D, dynamical simulations by TDDFT with superfluidity

 TDSLDA equations (or TDHFB, TD-BdG)

a vortex line exists here

 Computational details

Nuclear impurity:# of quasi-particle w.f.
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Method - TDSLDA
We performed 3D, dynamical simulations by TDDFT with superfluidity

 TDSLDA equations (or TDHFB, TD-BdG)

TITAN, Oak Ridge

 Computational details

NERSC Edison, Berkeley

MPI+GPU→ 48h w/ 200GPUs
for 10,000 fm/c

Nuclear impurity:# of quasi-particle w.f.
HA-PACS, Tsukuba
K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

How to extract the force
We directly measure the force

in dynamical simulation

 Newton’s law

if We keep a nuclear motion in a constant velocity
Superfluid neutronsvortexK. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Results of TDSLDA calculation:

Blue line: vortex-core

Red dot: c.m. of protons

v0: along x-axis
Force exerts on the nucleus
will be shown by a black arrowor

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021?

Results of TDSLDA calculation:“Repulsive”!!

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

Results of TDSLDA calculation:

Pinned configuration is dynamically unstable

K. Sekizawa

Lecture 3: Neutron-star “glitch” and neutron superfluidSun., Aug. 8, 2021

“Unpinned configuration”

“Pinned configuration”

Our goal and strategy
Goal: Unveil the mechanism of glitches
New collaboration started:
Nicolaus Copernicus Astronomical Centre
B. Haskell et al.

104m

Macroscopic➢ observations➢ hydrodynamics~10-10m

Mesoscopic➢ dynamics of vortices in a lattice of nuclei

Provide model ingredients
10-15-10-13m

Microscopic(e.g. filament model)

Nuclear Physics!!

➢ vortex-nucleus dynamics from neutrons and protons

Ongoing project - I
Mesoscopic simulation of pinning force with the Vortex Filament Model (VFM)➢ Even for a repulsive interaction, vortices could be immobilized by the lattice
Simulations by K. Kobczewski (a PhD student at WUT)

Talk by K. Kobczewski at POLNS18, March 26-28, 2018: https://indico.camk.edu.pl/event/10/contribution/8

Very preliminary

Ongoing project - II
Microscopic simulation of vortex dynamics with Gross-Pitaevskii Equation (GPE)➢ Vortices could be bent, and dynamics might be more complicated.
Simulation by T. Sasaki (a Master’s course student at Niigata University)

Very preliminary

Ongoing project - III
Microscopic simulation of Donnelly-Glaberson instability within TDSLDA➢ Kelvin waves are amplified, which may cause a turbulent state
Simulation by D. Pecak (a postdoc at WUT)
Normalflow

Box: 903 fm3 (Δx=1.5 fm)
EDF: BSk31

Very preliminarySummary

I hope you enjoyed our exciting adventure!

with a magic of DFT and TDDFT..

“Flat Earth” by iStock; “Fanciful view of ship sailing over edge of Earth” by Georgia Studies Images; “Star Wars Spaceship” by Valérian Pierret @Artstation

The transdisciplinary character is
one of the fascinating points of Nuclear Physics

What I showed in my lectures are only a tiny part of the huge field! :)Takeaway message
Sun(~109m)Earth(~107m)Solar system(~1021m)
Universe
Quark(<10-19m)
Daily life
(~1m)

u
u

Nucleus(~10-14m)
Atom (~10-10m)dProton(~10-15m)

Takeaway message
Sun(~109m)Earth(~107m)Nuclear physics is ubiquitous:
It Solarhas system
highly transdisciplinary character,(~1021m)

connecting different scales of physics in the Universe!
Universe

Quark(<10-19m)
Daily life
(~1m)

u
u

Nucleus(~10-14m)
Atom (~10-10m)dProton(~10-15m)

Kazuyuki Sekizawa
Associate Professor
Department of Physics, School of Science
Tokyo Institute of Technology
2-12-1 O-Okayama, Meguro, Tokyo 152-8551, Japansekizawa @ phys.titech.ac.jp
http://sekizawa.fizyka.pw.edu.pl/english/