YONUPA “Young Nuclear and Particle Physicist Group of Japan”
Summer School 2021: August 6-10, 2021 (via Zoom)
8/7 (Sat) 10:00-11:30 Sekizawa - Lecture 1

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

Brief personal history
1988: Born in Tokyo, grew up in Satte City in Saitama
2003-2006: Inagakuen Public High School
2006-2010: Tokyo University of Science (BSc)
2010-2015: University of Tsukuba (MSc-PhD)
Apr. 2015-Aug. 2017: Warsaw University of Technology, Poland (Postdoc)
Sep. 2017-Dec. 2017:

University of Washington, USA (Postdoc)Jan. 2018-Mar. 2021: Niigata University (Assistant Professor, tenure track)
Apr. 2021-Present: Tokyo Institute of Technology (Associate Professor)
Main research field: Nuclear Theory

Cherry blossoms
in Satte City

Lecture PlanTheory (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 PlanTheory (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

From quarks to atomic nuclei

Standard model of the elementary particles✓ Elementary particles: fundamental particles without structure✓ Four forces: strong, weak, electromagnetic, and gravitational forces✓ Particle physics explores an ultimate theory of the Universe

Higgs boson

Fermions

Gauge bosons

2nd gen.

Quarks

1st gen.

charm

electron neutrino

Leptons

EM forcebottomτ

Photons
Weak force

taumuonνeb

μeelectronGluonssstrange

Strong forcet
topddown

Chargecuup

3rd gen.

νμντmuon neutrino

tau neutrino

W±, Z0 bosons

From quarks to hadrons
Temperature

The QCD phase diagram✓ Hadrons: composite particles of quarks

Early universe

Baryonsu✓ Exploring the evolution of theUniverse through high-energy
nuclear experimentsu

Quark Gluon Plasma (QGP)

dudE.g. Proton

dNeutron➢ A nucleon (proton or neutron)
is a baryon.

MesonsHadronsu✓ Quark matter in NS core?
→ color superconductivity
AtomicNuclei

Neutron Stars

Density
Left figure: https://www.bnl.gov/newsroom/news.php?a=24281

E.g.

dPion (π+)

Energy scales
anddegrees of
freedom

Figure: https://www.asc.ohio-state.edu/physics/ntg/6805/slides/6805_overview_slides.php#Dofs

Not this “high energy” !!

Animation: Brookhaven National Laboratory (http://www.bnl.gov/rhic/); https://youtu.be/Vyq_AYWctSoWe collide two nuclei “gently”
and study quantum many-body dynamics of neutrons and protons
RCoulomb part?

Total~
Nuclear partIn “low-energy” nuclear physics,
we treat neutrons and protons as building blocks
What we study is:
A quantum many-body problem of fermions interacting through the nuclear force

Repulsive core
NucleonsuuddProtonud

 complex state dependence spin-orbit interaction
 tensor interaction ...

Neutron

Short-range attraction

Mean-field theories in nuclear physics

Single-particle vs. collective pictures
The Nobel Prize in Physics 1963

M.G. Mayer

J.H.D. Jensen

Shell model
Independent particle picture

Bethe-Weizsacker
mass formula
Liquiddrop modelstrongly correlated (collective) picture

How are they unified??

Photos: https://www.nobelprize.org; B/A (九州大学名誉教授 高田健次郎氏ウェブサイトより)
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

Hartree-Fock (HF) theory
Many-body wave function is approximated by a single Slater determinant
Unrestricted variation

HF approximation

Schrödinger equation:

Hartree-Fock equation:

whereSingle-particle Hamiltonian:Hartree potential (direct term):

Fock potential (exchange term):

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

How can we describe the pairing properties?

→ One may build the BCS theory on top of the Hartree-Fock― the HF+BCS approach ―Necessity of pairing✓ Soon after the establishment of the BCS theory, pairing was recognized in nuclear physics
The BCS wave function:

Phys. Rev. 108, 1175 (1957)

Phys. Rev. 110, 936 (1958)

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

Properties of the BCS wave function
We will just use:

NormWe can reorder the a† as there are 0 or even numbers of avanish by acting on the bra/ket
vanish by acting on the ket✓ Norm of the BCS wave function:➢K. Sekizawa

must be fulfilled for all k to normalize the BCS wave function!
Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

Properties of the BCS wave function
Particle number operator:

Particle-number expectation value

index = ka† acts on the bra

index = ka acts on the ketvanish by acting on the bra/ket

From each of two terms, one finds a unity(others vanish acting on the bra/ket)✓ Expectation value of the particle-number operator:➢K. Sekizawa

can be interpreted as a probability for a pairbeing occupied.
( is the probability being unoccupied.)
Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

Properties of the BCS wave function
Particle number operator:

Particle-number dispersion
The dispersion of the particle-number expectation value is defined by:

In a very similar manner, one can calculate (try it!):

Since we have

one finds:✓ Particle-number dispersion in the BCS state:➢ Particle-number fluctuation originates from the fractionally-occupied states around the Fermi level.
(where)K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

The BCS theory
Energy expectation value and its variation
Let us consider a simple Hamiltonian with a constant pairing interaction:

The energy expectation value reads (try it!):Taking a variation with respect to vk , one obtains:

The BCS condition
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

The BCS theory✓ As a result of the variation,

Single-particle energy:, we got the BCS condition:
Pairing gap:

One can analytically solve it forand, leading to:✓ These amplitudes satisfy★
the BCS condition(i.e.,)

Since we haveone obtains:➢ The gap equation:
where λ can be determined by: Particle-number condition➢ Since the above set of equations sustains the relation “★”, which is equivalent to the BCS condition,
it offers a practical way of determining Δ, λ, and vk, for a given set of εk.
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

More sophisticated approach to include pairing→ the Hartree-Fock-Bogoliubov theory(a.k.a. the Bogoliubov-de Gennes theory)

The HFB theory➢ The BCS wave function can be interpreted as a vacuum of quasi-particles, i.e.:
i.e.

whereAnnihilation of a quasiparticle in a state kCreation of a quasiparticle in a state -k➢ A quasiparticle is a mixture ofparticle and hole states➢ The HFB wave function would be the most generalized mixture of particle and hole states:i.e.

✓ The “generalized” Bogoliubov transformation is given by:

or in a matrix form:

where there holdK. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

The HFB theory
Energy expectation value: Density matrix: Mean field: Pairing tensor: Single-particle Hamiltonian

Result of a variation

HFB matrix:: Pairing field

The HFB condition
Generalized density matrix:

or equivalently➢ By diagonalizing the HFB matrix, one can determine U and V, which fulfill the HFB condition!
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

(TD)DFT in a tiny nutshell

DFTWhat is DFT?
A theory which gives us access to the exact solution
Equivalent!
(for a special EDF)
Kohn-Sham equation

EDF

This is the key!

Energy can also be written as a functional of density

w.f. is a functional of density
P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964)
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

What is DFT?
A theory which gives us access to the exact solution
Equivalent!
(for a special EDF)
Kohn-Sham equation

EDF

This is the key!

Existence was proven, but “shape” is unknown :(K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

Great Success of the Density Functional Theory
Number of publications with “DFT” (till 2013)

The Nobel Prize in Chemistry 1998

Walter Kohn

John Pople ©https://www.nobelprize.org
Year
A. Galano and J.R. Alvarez-Idadoy, J. Compt. Chem. 35, 2019 (2014)

Fullerene: C60

Si crystal

C-Z. Gao et al.,J. Phys. B: At. Mol. Opt. Phys. 48, 105102 (2015)

Y. Shinohara, K. Yabana, Y. Kawashita, J.-I. Iwata, T. Otobe, and G. F. Bertsch,
Phys. Rev. B 82, 155110 (2010)

The seminal papers on DFT

P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)
W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965)

19,015 citations!
24,384 citations!

What’s behind the wonder of DFT?
→ Hohenberg-Kohn theorem & Kohn-Sham scheme

DFT in a tiny nutshell (1/2) - Hohenberg-Kohn theorem
Assumption:
Two different wave functions
andprovide the same one-body density

Now, one finds:

・・・ (a)However, one also gets:・・・ (b)(a) + (b)?
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

DFT in a tiny nutshell (1/2) - Hohenberg-Kohn theorem
Assumption:
Two different wave functions Our
andassumption was wrong!
provide the same one-body density

Now, one finds:“reductio ad absurdum”

・・・ (a)However, one also gets:・・・ (b)(a) + (b)?
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

DFT in a tiny nutshell (1/2) - Hohenberg-Kohn theorem
Assumption:

The 1st HK theorem:

Two different wave functions
andThereisaone-to-one
provide the same one-body density correspondence between Ψ and ρ

Now, one finds:“reductio ad absurdum”

・・・ (a)However, one also gets:・・・ (b)(a) + (b)?
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

DFT in a tiny nutshell (1/2) - Hohenberg-Kohn theorem
The 1st HK theorem:

There is a one-to-one correspondence between Ψ and ρ

The one-body density can be the fundamental degree of freedom!
HKSE

The 2nd HK theorem:Ψ can be written as a functional of ρ:
Any observable can be written as a functional of ρ:

Minimization of EDF with respect to ρ will provide Eg.s. and ρg.s.:: Energy Density Functional (EDF)

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

DFT in a tiny nutshell (2/2) - Kohn-Sham scheme
Kohn-Sham scheme makes DFT computationally solvable
W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965)

Kohn-Sham scheme:✓ The EDF of the many-body system:

This contains all necessary info to get the many-body ground state✓ Let us rewrite it as:
E’ is the rest of EDF except the one-body kinetic term

✓ Taking the variation w.r.t. single-particle orbitals,one finds:: Kohn-Sham equation
Kohn-Sham potential:

K. Sekizawa

One-body density:

Kohn-Sham orbitals:

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

Then, how it works?

DFT in Nuclear Physics

Proton number

All nuclei can be described with a single EDF

N, Z

Neutron number
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

DFT in Nuclear Physics
All nuclei can be described with a single EDF

Proton number

Neutron star
N, Z~~
Neutron number
K. SekizawaLecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

What is TDDFT?
TDDFT is a time-dependent extension of DFTTDSE for an N-body system

for all timett0Time

TDKS eq. in TDDFT

E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984); R. van Leeuwen, Phys. Rev. Lett. 82, 3863 (1999).
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

TDDFT in a tiny nutshell - Runge-Gross theorem
E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984)

One-body current density:

One-body density:

TDSE:

The rate of change of the current density reads:

・・・ (a)・・・ (b)・・・ (★)(a)－(b)*One can obtain equations for higher-order derivatives in the same manner

As in the case of static DFT, we have a TDKS scheme
according to the following one-to-one correspondences:
TDSE

RG

*Taking divergence of Eq.(★) and using the continuity equation, one can obtain equations for the density.
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

TDDFT in Nuclear Physics
TDDFT is a versatile tool!!
IVGDR

Vortex-nucleus dynamics

Phys. Rev. Lett. 117, 232701 (2016)
G. Wlazłowski, K.S., P. Magierski, A. Bulgac, and M.M. Forbes
Phys. Rev. C 84, 051309(R) (2011)
I. Stetcu, A. Bulgac, P. Magierski, and K.J. Roche

Induced fission of 240Pu

Phys. Rev. Lett. 116, 122504 (2016)
A. Bulgac, P. Magierski, K.J. Roche, and I. Stetcu
K. Sekizawa

Low-energy heavy-ion reactions
TimePhys. Rev. Lett. 119, 042501 (2017)
P. Magierski, K.S., and G. Wlazłowski

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFTSat., Aug. 7, 2021

Q. What are difference and relevance between
Hartree-Fock theory and DFT?

A. HF ≠ DFT, but HF with an “effective interaction”
may be regarded as Kohn-Sham DFTHartree-Fock theory vs. DFTRegarding ESkyrme as an EDF, Skyrme-HF eq. may be regarded as KS eq. in DFT
Unrestricted variation
Schrödinger eq:

Unrestricted Variationso, of course, HF ≠ DFT

Variation for a Slater determinantHartree-Fock eq:*Integro-differential eq.

Slater determinant:Orthonormal condition:

Hartree potential (direct term):

Fock potential (exchange term):

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Variation
for a Slater det.

Sat., Aug. 7, 2021

Hartree-Fock theory vs. DFTRegarding ESkyrme as an EDF, Skyrme-HF eq. may be regarded as KS eq. in DFT➢ One may also regard Skyrme-TDHF as TDKS in TDDFT
Unrestricted variation
Schrödinger eq:

Unrestricted Variation
Kohn-Sham equation:

Variation
for a Slater det.

Variation for a Slater determinantHartree-Fock eq:

Skyrme “effective interaction”*Integro-differential eq.

*t0-3, x0-3, W0, and α are the parameters

T.H.R. Skyrme, Philos. Mag. 1, 1043 (1956): Relative coordinate: C.M. coordinate: Relative momentum op.
: Spin exchange op.

Skyrme-Hartree-Fock eq:*E is a functional of various densities*ξα stands here for various densities
K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

SummarySummary of Lecture 1
Takeaway message✓ (TD)DFT is a powerful theoretical framework to study
quantum many-body problems of nucleons at low energy
from nuclei to neutron stars

K. Sekizawa

Lecture 1: An introduction to microscopic mean-field approaches and (TD)DFT

Sat., Aug. 7, 2021

Lecture PlanTheory#1. An introduction to microscopic mean-field approaches and (TD)DFT

Nuclear reactions#2. Recent advances in microscopic approaches for heavy-ion reactions
Superheavy element synthesis & deep-inelastic collisions

Neutron stars#3. Neutron-star “glitch” and neutron superfluid
Dynamics of quantized vortices in the inner crust of neutron stars

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/