Haozhao LIANG

Haozhao LIANG ^{（梁豪兆）}

RIKEN Nishina Center, Japan

Graduate School of Science, the University of Tokyo, Japan

November 11, 2017

Workshop of Recent Developments in QCD and QFT November 9-12, 2017, NTU, Taipei

Density Functional Theory with uncertainty quantification from Functional Renormalization Group in Kohn-Sham scheme

In collaboration with

Tetsuo Hatsuda (RIKEN, Japan) and Yifei Niu (ELI-NP, Romania)

http://www.nishina.riken.go.jp/index_e.html

Chart of Nuclei (2017)

~ 3000 nuclei

stable nuclei ~ 300 nuclei

unstable nuclei observed so far ~ 3000 nuclei

drip-lines (limit of existence)（theoretical predictions) ~ 6000 nuclei magic numbers

1985年の核図表

~ 1500 nuclei

(1985)

Nuclear chart

Radioactive isotope beam facilities

NSCL (USA) FRIB (2020)

GANIL (France) SPIRAL2, Phase 2 (2016-18)

ISOLDE, CERN HIE-ISOLDE (2014) GSI (Germany) FAIR (2018-20)

RIBF RIKEN (Japan) SRC+BigRIPS (2007) EURISOL (Europe)

(Planned) TRIUMF

ISAC I-II

RISP (Korea) (Funded)

Existing Upgrading

Best in the world

~70 % speed of light !

Atomic nuclei

Atomic nucleus is a rich system in physics

 quantum system

 many-body system (A ~ 100, spin & isospin d.o.f.)

 finite system (surface, skin, halo, …)

 open system (resonance, continuum, decay, …)

Spin

Isospin

Tanihata:1985

Neutron halos R ~ A^1/3? Not always!

11Li: a size as ²⁰⁸Pb

Physics of exotic nuclei

Weakly-bound

Threshold Continuum Open Q.S.

Halo

Deformation

Shape decoupling

Halo Large spatial ext.

Low-density N.M.

2N correlation Shell evolution

New radioact.

1p emission 2p emission 2n emission Clustering

Prof. Shan-Gui Zhou’s plenary talk

@ INPC2016, Australia

State-of-the-art nuclear methodologies

http://www.unedf.org/

 Density functional theory (DFT) aims at understanding both ground- state and excited-state properties of thousands of nuclei in a consistent and predictive way.

Covariant density functional theory

Covariant density functional theory (CDFT)

 Fundamental: Kohn-Sham Density Functional Theory

 Scheme: Yukawa meson-exchange nuclear interactions

Comparing to traditional non-relativistic DFT

 Effective Lagrangian

connections to underlying theories, QCD at low energy

 Dirac equation

consistent treatment of spin d.o.f. & nuclear saturation properties (3-body effect)

 Lorentz covariant symmetry

unification of time-even and time-odd components

Aoki et al., Prog. Theor. Exp. Phys. 2012, 01A105 (2012) Nobel Prize 1949 Nobel Prize 1998

Dirac and RPA equations

 Energy functional of the system

 Dirac equations for the ground-state properties

with

 RPA equations for the vibrational excitation properties

 δE/δρ  equation of motion for nucleons: Dirac (-Bogoliubov) equations

 δ²E/δρ²  linear response equation: (Q)RPA equations

Ring & Schuck, The Nuclear Many-Body Problem (Springer, New York, 1980)

Nuclear mass models

Accuracy for ²⁰⁸Pb: ~1/1600

Gamow-Teller resonances

CDFT+RPA for Gamow-Teller resonances (ΔS = 1, ΔL = 0, J^π = 1⁺)

HZL, Giai, Meng, Phys. Rev. Lett. 101, 122502 (2008)

 GTR excitation energies can be reproduced in a fully self-consistent way.

 New and most important ingredient:

Fock terms in CDFT

exp. @ RCNP

r -process nucleosynthesis & nuclear β decays

 Nuclear masses  path of r-process

 Nuclear β-decay rates  timescale of r-process

 EURICA project is providing lots of new β-decay data towards r-process path.

Courtesy of S. Wanajo

The 11 greatest unanswered questions of physics

Key exp. @ RIKEN

Question 3

How were the heavy elements from iron to uranium made?

Rapid neutron-capture process (r-process)

β decays and r -process

Nuclear β-decay rates and r-process flow (Z = 20 ~ 50 region)

Niu, Niu, HZL, Long, Niksic, Vretenar, Meng, Phys. Lett. B 723, 172 (2013)

FRDM+QRPA: widely used nuclear input RHFB+QRPA: our results

 Classical r-process calculation shows a faster r-matter flow at the N = 82 region and higher r-process abundances of elements with A ~ 140.

“There exist at least three families of

quarks in nature.”

"Only three?”

CKM matrix and its unitarity test

Plenary talk in INPC2010 “Precision Electroweak Tests of the Standard Model” by Professor William Marciano

Nobel Prize 2008

Cabibbo-Kobayashi-Maskawa matrix

Cabibbo-Kobayashi-Maskawa matrix

 quark eigenstates of weak interaction quark mass eigenstates

 unitarity of CKM matrix test of Standard Model

Unitarity test Particle Data Group 2016

 the most precise test comes from |V_ud|² + |V_us|² + |V_ub|²

 the most precise |V_ud| comes from nuclear 0⁺  0⁺ superallowed β transitions Nuclear superallowed β transitions

 experimental measurements

 theoretical corrections (isospin symmetry-breaking corrections)

CKM matrix and its unitarity test

Isospin corrections & V

Isospin corrections by self-consistent CDFT cited by PDG 2010, 2012, 2014, ...

HZL, Giai, Meng, PRC 79, 064316 (2009); Satula et al., PRL 106, 132502 (2011); PRC 86, 054316 (2012)

 To our best knowledge: |V_ud|² + |V_us|² + |V_ub|²: 0.997 ~ 1.000 (the 4^th family?)

 ongoing studies ……

A dream for next-generation DFT

quantum-field-theory oriented DFT

also cf.

Schwenk & Polonyi, arXiv:0403011 [nucl-th]

Kutzelnigg, JMS 768, 163 (2006) Drut, Furnstahl, Platter, PPNP 64, 120 (2010) Braun, JPG 39, 033001 (2012) Metzner et al., RMP 84, 299 (2012) Drews & Weise, PPNP 93, 69 (2017)

……

EDF from effective action F_HK[ρ] ~ Γ[ρ]/β

(Legendre transform)

theoretical uncertainties

from EFT Γ⁽²⁾, Γ⁽³⁾, Γ⁽⁴⁾ …

(power counting)

non-perturbative nature by

renormalization group

∂_kΓ_k[ρ] = Tr{…}

(flow eq.)

Interdisciplinary:

(lattice) QCD hadron cold atom condensed matter quantum chemistry

……

IUPAP Young Scientist Prize

@ INPC2016, Australia

Density Functional Theory

Hohenberg-Kohn theorem [Phys. Rev. 136, B864 (1964)]

 There exist a universal density functional F_HK[ρ(x)].

 The ground-state energy E_gs attains its minimum value when the density ρ(x) has its correct ground-state value.

The aim of density functional theory (DFT) is

 to reduce the many-body quantum mechanical problem formulated in terms of N-particle wave functions Ψ to the one-particle level with the local density distribution ρ(x).

 HK variational principle

Where is a universal functional, which is valid for any number of particles N and for any external field U(x).

Goal: F

[ ρ]

EDF from effective action

Strategy: F

[ ρ]  Γ[ρ]  partition function  path integral

 Classical action in Euclidean space

where U(x) is one-body potential and V(x₁, x₂) is two-body interaction.

 Partition function in two-particle point-irreducible (2PPI) scheme

external source J couples ψ^ψ at the same space-time.

 Thermodynamic potential / generating function / Schwinger function

 Local density

 Effective action  Legendre transform of W with respect to J

Connection to Hohenberg-Kohn theorem

 The universality of the Hohenberg-Kohn functional F_HK[ρ] follows from the fact that the background U potential can be absorbed into the source terms J by a simple shift J  J – U.

proof

 Lippmann-Schwinger eq. / Bethe-Goldstone eq. / Brueckner theory Brueckner Hartree-Fock, hole-line expansion … (in 1960s, 70s)

 Functional Renormalization Group (FRG) --- Wetterich, PLB 301, 90 (1993)

Flow equation

Cited 1500+ times (google scholar, October 2017)

in QCD, hadron, nuclear, cold atom, condensed matter, quantum chemistry ……

Non-perturbative nature of interaction

Relativistic BHF for finite nuclei

Shen, Hu, HZL, Meng, Ring, Zhang, Chin. Phys. Lett. 33, 102103 (2016) Shen, HZL, Meng, Ring, Zhang, PRC 96, 014316 (2017)

 Lippmann-Schwinger eq. / Bethe-Goldstone eq. / Brueckner theory Brueckner Hartree-Fock, hole-line expansion … (in 1960s, 70s)

 Functional Renormalization Group (FRG) --- Wetterich, PLB 301, 90 (1993)

Non-perturbative nature of interaction

 FRG + DFT --- Schwenk & Polonyi, arXiv:0403011 [nucl-th]

Flow equation

Relativistic BHF for finite nuclei

Shen, Hu, HZL, Meng, Ring, Zhang, Chin. Phys. Lett. 33, 102103 (2016) Shen, HZL, Meng, Ring, Zhang, PRC 96, 014316 (2017)

 FRG Flow equation: a set of coupled differential equations (infinite hierarchy)

Theoretical uncertainty

 Proper power counting by Γ⁽⁰⁾, Γ⁽²⁾, Γ⁽³⁾, Γ⁽⁴⁾ …？

 Controllable theoretical uncertainty?

… …

To solve exactly?

To estimate smartly?

Model setup

 Classical action (0D in space-time, bosonic d.o.f.)

 Partition function (2PPI scheme)

Exact solution

 Ground-state energy (v₀ = v/ω⁴)

 Ground-state density

φ

-theory in zero dimension

Typical cases

v/ω⁴ = 0.01 v/ω⁴ = 1 v/ω⁴ = 100

perturbative non-perturbative highly non-perturbative

2% error in 6^th-order cal.

barely discussed

Keitel & Bartosch, JPA 45, 105401 (2012) Kemler & Braun, JPG 40, 085105 (2013)

 FRG + DFT Flow equation

 Optimized expansion

 Non-interacting part

 Correlation part

Our ideas

Ideas of Kohn-Sham

 To introduce an artificial non-interacting system which provides the same ground-state density ρ_gs with a Kohn-Sham (mean-field) potential

 Difference between interacting and non-interacting systems is absorbed in the correlation (beyond-mean-field) part of EDF, E_x[ρ].

HZL, Niu, Hatsuda, arXiv:1710.00650

 Coupled differential equations  Optimized FRG + DFT

Optimized FRG + DFT

Estimating upper/lower bounds

… …

To solve

Keeping as

KS counterparts

 Perturbative case v₀ = v/ω⁴ = 0.01

Typical cases (I)

 1^st-order optimized FRG result is on top of the exact solution in a very large density region.

 Theoretical uncertainty is invisible in the figure.

 Effective action vs density, F_HK[ρ]. HZL, Niu, Hatsuda, arXiv:1710.00650

 Non-perturbative case v₀ = v/ω⁴ = 1

Typical cases (II)

 1^st-order result reproduces the exact solution in a wide density region, with corresponding uncertainty.

 2^nd-order result does improve.

Kemler & Braun (2013)

 Effective action vs density, F_HK[ρ].

KS optimized FRG

 Highly non-perturbative case v₀ = v/ω⁴ = 100

Typical cases (III)

 1^st-order result is still able to describe the exact solution with large uncertainty.

 2^nd-order and 3^rd-order results improve step by step

— accuracy of ρ_gs increases by factor 4, E_gs by factor 10 at each order.

 Ground-state density and energy.

In summary

 Ground-state densities and energies: φ⁴-theory in 0D

from φ

-theory in 0D to 3+1D finite nuclei … DFT  ideas of QFT (effective action + RG + EFT)

 EDF F_HK[ρ] is derived from effective action Γ[ρ] in 2PPI scheme with Legendre transform.

 Non-perturbative nature of interaction is handled by FRG with flow equation.

 Beyond-mean-field effects γ⁽ⁿ⁾ in F_HK[ρ] are taken into account order-by- order with (proper) theoretical uncertainties.

8^th digit 0.08% 0.8%

HZL, Niu, Hatsuda, arXiv:1710.00650

Acknowledgments

RIKEN Takumi Doi, Tetsuo Hatsuda, Pascal Naidon, Zhongming Niu, Hiroyuki Sagawa, Masaki Sasano, Kazuko Tanabe ……

U. Tokyo Asahi Chikaoka, Tomoya Naito, Daisuke Ohashi INFN-Milan Gianluca Colò, Xavier Roca-Maza

Anhui U. Jian-You Guo, Min Shi

Tohoku U. Kouichi Hagino, Yusuke Tanimura IBS-Raon Youngman Kim, Yeunhwan Lim Lanzhou U. Wen Hui Long

Peking U. Jie Meng, Shihang Shen

JAEA Futoshi Minato

Tsukuba U. Takashi Nakatsukasa, Zhiheng Wang Zagreb U. Tamara Nikšić, Dario Vretenar

ELI-NP Yifei Niu

TU München Peter Ring

IPN-Orsay Nguyen Van Giai Osaka U. Hiroshi Toki Sophia U. Shinya Wanajo Argonne Pengwei Zhao

ITP-CAS Shan-Gui Zhou

Thank you!