1. Relativistic Brueckner-Hartree-Fock Theory for finite nucleus
CRC110 Workshop on
Nuclear Dynamics and Threshold Phenomena
Ruhr-Universität Bochum, April 5-7, 2017
Relativistic Brueckner-Hartree-Fock Theory for finite nucleus
Jie MENG （孟 杰）
School of Physics，Peking University（北京大学物理学院）

2. Outline Introduction Full Relativistic Brueckner-Hartree-Fock
Calculation for 4He, 16O, 40Ca, and 48Ca
Summary & Perspectives
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3. Nuclear Energy Density Functional
Nuclear Energy Density Functionals: the many-body problem is mapped onto a one-body problem without explicitly involving inter-nucleon interactions!
Kohn-Sham Density Functional Theory
For any interacting system, there exists a local single-particle potential h(r), such that the exact ground-state density of the interacting system can be reproduced by non-interacting particles moving in this local potential.
The basic idea of NEDFT is mapping the many-body problem onto a one-body problem without …. In fact, we borrowed this idea from the Kohn-Sham DFT for electron systems. The KS DFT told us that …. How to determine this local potential? It can be obtained according to the Hohenberg-Kohn theorem by a variation of the energy functional with respect to the density. So, the practical usefulness of the Kohn-Sham scheme depends entirely on whether Accurate Energy Density Functional can be found!
The practical usefulness of the Kohn-Sham scheme depends entirely on whether Accurate Energy Density Functional can be found!
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4. Density functional theory in nuclei
Nuclear DFT has been introduced by effective Hamiltonians: by Vautherin and Brink (1972) using the Skyrme model as a vehicle
Based on the philosophy of Bethe, Goldstone, and Brueckner one has a density dependent interaction in the nuclear interior G(ρ)
At present, the ansatz for E(ρ) is phenomenological:
Skyrme: non-relativistic, zero range
Gogny: non-relativistic, finite range (Gaussian)
CDFT: Covariant density functional theory
14:33

5. Why Covariant? Spin-orbit automatically included
P. Ring Physica Scripta, T150, (2012)
Spin-orbit automatically included
Lorentz covariance restricts parameters
Pseudo-spin Symmetry
Connection to QCD: big V/S ~ ±400 MeV
Consistent treatment of time-odd fields
Relativistic saturation mechanism …
Pseudospin symmetry
Now, we come to this question why a functional should be covariant? There are some advantages for the Covariant density functional theory. For example, because it includes the spin degree of freedom naturally, it explains naturally the large spin-orbit splitting and pseudo-spin symmetry. There is a great success of rela. Brueckner calculations for the nuclear matter. One can treat the time-odd fields consistently which are important to nuclear rotations. Later we will see, we have two large fields of vector and scalar potentials, and this provides the relativistic saturation mechanism.
Hecht & Adler
NPA137(1969)129
Arima, Harvey & Shimizu
PLB 30(1969)517
Ginocchio PRL 78, 436
Brockmann & Machleidt, PRC42, 1965 (1990)

6. The relativistic mean field (RMF) theory, based on either the finite-range meson-exchange or point-coupling interactions, has received much attention due to its successful description of numerous nuclear phenomena in stable as well as exotic nuclei
International Review of Nuclear Physics (Vol 10)
Relativistic Density Functional for Nuclear Structure
World Scientific(Singapore (2016)
B. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1.
P.-G. Reinhard, Rep. Prog. Phys. 52 (1989) 439
P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193.
D. Vretenar, A. Afanasjev, G. Lalazissis, P. Ring, Phys. Rep. 409 (2005) 101.
J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S. Geng, Prog. Part. Nucl. Phys. 57 (2006) 470.

7. Meson-exchange version of CDFT
CDFT：Relativistic quantum many-body theory based on DFT and effective field theory for strong interaction
Strong force: Meson-exchange of the nuclear force
(Jp T)=(0+0) s w r
(Jp T)=(1-0)
(Jp T)=(1-1)
Sigma-meson:
attractive scalar field
Omega-meson:
Short-range repulsive
Rho-meson:
Isovector field
Electromagnetic force: The photon

8. Point-coupling version of CDFT
Elementary building blocks
Energy Density Functional
Densities and currents
Isoscalar-scalar
Isoscalar-vector
Isovector-scalar
Isovector-vector
The elementary building blocks for a point-coupling density functional could have such 10 terms. At the present, we build our density functional with the densities and currents coming from only the most important 4 channels including the isocalar-scalar, …. Finally, our energy density functional has such a form, it contains the kinetic term, 2nd order term, high order term, derivative term, and the electromagnetic term. As will see later, the coupling constants in this functional would be determined by fitting to the properties of finite nuclei.

9. Covariant density functional
Meson Exchange
Point Coupling
Nonlinear parameterizations:
Nonlinear parameterizations:
NL3, NLSH, TM1, TM2, PK1, …
PC-LA, PC-F1, PC-PK1 …
Density dependent parameterizations:
Density dependent parameterizations:
TW99, DD-ME1, DD-ME2, PKDD, …
DD-PC1, … 2
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10. Covariant density functional PC-PK1
Zhao, Li, Yao, Meng, PRC 82, (2010)
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11. CDFT for Nuclear matter
Saturation properties:

12. Kτ = Kτ,v + Kτ,s A-1/3
Data from from Umesh Garg, also H. Sagawa et al., Phys. Rev. C 76, (2007)
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13. CDFT: Meson-exchange/Point Coupling
Very successful in nuclear physics
Ring PPNP1996, Vretenar Phys.Rep.2005,
Meng PPNP2006
Spin-orbit splitting
Pseudo-spin symmetry
Nuclear saturation properties
Exotic nuclei
Excellent reproduction of nuclear properties ……
Meng, Peng, Zhang, Zhao, Front. Phys., 8(2013) 55–79
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14. The relativistic Hartree-Fock theory
Bouyssy PRC Bernardos PRC 1993 Marcos JPG Bürvenich PRC 2002
In addition of the RMF advantages, there are
Pion contribution included
the contributions due to the exchange (Fock) terms
the pseudo-vector π-meson.
Nuclear effective mass
Fully self-consistent description for spin-isospin excitation ……
Long PLB2006, Long PRC2007, Liang PRL2008, Liang PRC2009 14

15. DDRHF(B): pion and Exchange term
Long, Giai, Meng, Physics letters B640 (2006) 150
Long, Ring. Giai, Meng, PHYSICAL REVIEW C81 (2012)
Long, Ring. Giai, Bertulani, Meng, PHYSICAL REVIEWC 81(2010)
RHFB equation
Single particle Hamiltonian:
Kinetic energy:
Local potentials:
Non-local Potentials:
Pairing Force: Gogny D1S
Dirac Woods-Saxon Basis: solve the integro-differential RHFB equation 15 15 15

16. Charge-exchange excitation modes
No contribution from isoscalar mesons (σ,ω), because exchange terms are missing.
π-meson is dominant in this resonance.
zero-range pionic counter-term g’ has to be refitted to reproduce the data.
Isoscalar mesons (σ,ω) play an essential role via the exchange terms.
While, π-meson plays a minor role.
g’ = 1/3 is kept for self-consistency.
RH + RPA
RHF + RPA
Liang, Giai, Meng, PRL 101, (2008)
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17. Lcalized form of Fock terms
The fine structure of spin-dipole excitations in O-16 is reproduced quite well in a fully self-consistent RPA calculation based on the RHF theory
A localized form of Fock terms is proposed with considerable simplicity as compared to the conventional Fock terms.
Based on this localized RHF theory, the spin-dipole excitation in Zr-90 is well reproduced with a RPA calculation.
Liang, Giai, Meng PRL 101, (2008)
Liang, Zhao, Meng Phys. Rev. C 85, (2012)
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Liang, Zhao, Ring, Roca-Maza, Meng Phys. Rev. C 86, (2012)

18. Relativistic Brueckner-Hartree-Fock Theory for finite nucleus
ab initio calculation
Relativistic Brueckner-Hartree-Fock Theory for finite nucleus

19. ab initio calculation in nuclear physics
ab initio----- “from the beginning”
without additional assumptions
without additional parameters
ab initio in nuclear physics
with realistic nucleon-nucleon interaction
with some few-body methods and many-body methods, such as Monte Carlo method, shell model and energy density functional theory
BHF
ab initio in nuclear matter
Variational method Akmal PRC1998
Green’s function method Dickhoff PPNP2004
Chiral Perturbation theory Kaiser NPA2002
Brueckner-Hartree-Fock (BHF) theory Baldo RPP2012
Relativistic BHF (RBHF) theory Brockmann PRC1990
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20. ab initio calculation in nuclear physics
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ab initio calculation for light nuclei
Gaussian Expansion Method Hiyama PPNP2003
Green Function Monte Carlo Method Pieper PRC2004
Lattice Chiral Effective Field Theory Lee PPNP2009
No-Core Shell Model Barrett PPNP2012 ……
ab initio calculation for heavier nuclei
Coupled Channel method Hagen PRL2009
BHF theory Hjorth-Jensen Phys.Rep.1995
With HJ potential Dawson Ann.Phys.1962
With Reid potential Machleidt NPA1975
With Bonn potentials Muether PRC1990
在低能区域，仅需要考虑质量量纲不大于４的项，这也就是通常所说的可重整性。再加上一些需要满足的对称性，规范对称性，得到了数目有限的若干种相互作用的量子场论。
16O in BHF method in Bonn potential

21. ab initio calculation for nucleus
Relativistic Brueckner Hartree-Fock: nuclear matter
Nuclear matter Anastasio PRep 1978 Brockmann PLB 1984 ter Haar PRep. 1987
Defining an effective medium dependent meson-exchange interaction based upon the
nuclear matter G matrix Brockmann PRC1990 Brockmann PRL Fritz PRL 1993
ab initio attempt for finite nucleus: extracted interaction from the ab initio calculation for nuclear matter
Density-dependent relativistic mean field theory Brockmann PRL1992
Density-dependent relativistic Hartree-Fock theory Fritz PRL1993
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22. Full Relativistic Brueckner–Hartree–Fock
Ab initio calculations, the description of finite nuclei with a bare nucleon-nucleon (NN) interaction, form a central problem of theoretical nuclear physics.
Nonrelativistic potentials systematically failed to reproduce the saturation properties of infinite nuclear matter in ab initio calculations. It has been concluded that the bare three-body force plays an essential role.
Relativistic Brueckner–Hartree–Fock theory, based on RHF calculations with the 𝐺-matrix, can reproduce empirical saturation data without three-body forces.
Inspired by the success of the RBHF theory in nuclear matter, it is a natural extension to study finite nuclei in the same framework.
Shi-Hang Shen, Jin-Niu Hu,Hao-Zhao Liang, Jie Meng, Peter Ring, Shuang-Quan Zhang,
Relativistic Brueckner–Hartree–Fock Theory for Finite Nuclei .
Chin. Phys. Lett. 33 (2016)
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23. T-Matrix and G-Matrix Lippmann-Schwinger Equation
Bethe-Goldstone Equation
Lippmann Phys. Rev 1950
Brueckner Phys. Rev 1955
V is the realistic NN interaction
E is the incident energy
T-matrix is for two-body scattering
E is the starting energy
Q is the Pauli operator
G-matrix is for many-body problem
The corresponding EOS in HF
The corresponding EOS in HF
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24. Bethe-Goldstone Equation
K. A. Brueckner, C. A. Levinson, and H. M. Mahmoud, Phys. Rev. 95, 217 (1954)
Sum up ladder diagrams to infinite order
Bethe-Goldstone equation
W is the starting energy, its value depends on the position of G-matrix in the diagram.
εm , εn are the RHF single particle energies.
Q is the Pauli operator which forbids the states being scattered below Fermi surface.

25. Relativistic Brueckner-Hartree-Fock Theory
Relativistic Hartree-Fock equation in complete basis,
for details, e.g.
where D are the expansion coefficients:
RBHF total energy
together with exchange term,
W. Long, N. Van Giai, and J. Meng, PLB 640, 150 (2006)
m’ n’
a b a m n b a m n b
a b =
RBHF
RHF

26. Iteration of RBHF Solution
Initial single-particle basis trial for RBHF final solution
Bethe-Goldstone equation
Single-particle potential
RHF iteration
If converged , RBHF iteration finishes.
Basis transformation
Go back to step 2.
Solving with matrix inversion method
M. Haftel and F. Tabakin, NPA 158, 1 (1970) =

27. Single-Particle Potential in Nuclear Matter
Bethe-Brandow-Petschek (BBP) theorem: if the s.p. potential of hole states (k <= kF) defined in the following way, certain higher order diagrams will be canceled:
For s.p. potential of particle states (k > kF), there are several choices:
Continuous choice : same as hole states.
Gap choice : U = 0 for k > kF.
In between : ε’ = energy of hole states.
H. Bethe, B. Brandow, and A. Petschek, Phys. Rev. 129, 225 (1963)
R. Rajaraman and H. Bethe, Rev. Mod. Phys. 39, 745 (1967)

28. Single-Particle Potential in Finite Nuclei
In this work, ε’ is chosen in the range of energy of occupied states.
K. Davies, et al., Phys. Rev. 177, 1510 (1969)

29. Center-of-Mass Motion
Intrinsic Hamiltonian
with
Projection after variation (PAV)
Projection before variation (PBV)
Ref: P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer, 1980

30. RHF Check
Motivation: by repeating previous RHF calculation, we can check the correctness of
Two-body matrix elements.
RHF equation solving.
Numerical details:
Object: 16O.
Interaction: Bouyssy (a,b,c,e) A. Bouyssy, et al., PRC 36, 380 (1987).
Box size: R = 10 fm (step size 0.05 fm).
Basis space: lcut = 3; εDcut = MeV, εFcut = 300 MeV.
Convergence condition: max | Uij (new) – Uij (old) | < 10-4 fm-1.
Center-of-mass motion: No.

31. RHF Check
Binding energy per nucleon -E/A (MeV), charge radius rc (fm), and proton 1p spin-orbit splitting ΔLS (MeV) with our code (Shen), in comparison with Bouyssy A. Bouyssy, et al., PRC 36, 380 (1987) and results from Long’s DDRHF code W. Long, N. Van Giai, and J. Meng, PLB 640, 150 (2006).
RH: σ + ω
RHF: σ + ω
RHF: σ + ω + π
RHF: σ + ω + π + ρ
without Coulomb exchange
with Coulomb exchange
Correctness of RHF part is confirmed.

32. Numerical Details for RBHF Calculation
Object: 4He, 16O, 40Ca, 48Ca.
Interaction: Bonn A potential. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989)
Cut-offs for basis space: lcut = 20, εcut = 1100 MeV, εDcut = MeV.
Basis: fixed Dirac Woods-Saxon basis and self-consistent RHF basis.
Coulomb exchange term: relativistic local density approximation.
Center-of-mass correction: included in variation.
as in density functional SLy E. Chabanat, et al., NPA 635, 231 (1998).
Box size: R = 7 fm.
S. Zhou, J. Meng, and P. Ring, PRC 68, (2003)
H. Gu, et al., PRC 87, (2013)

33. Convergence to Energy Cut-Off
Total energy and charge radius of 16O as a function of energy cut-off.
Rough estimation of the dimension of |ab> :
(20 × 2 × 20)2 = 640,000 (symmetry) ≈ 60,000
l s n
Storage: 256 GB CPU time: 224 h
Intel(R) Xeon(R) CPU E GHz
Satisfactory convergence is achieved near εcut = 1.1 GeV.

34. Center-of-mass Motion
Intrinsic Hamiltonian
Include -Hcm in RBHF iteration (PBV) or c.m. correction after RBHF iteration (PAV).
Total energy E and energy of c.m. motion Ecm at each iteration step.
Δ ≈ 9 MeV
Δ ≈ 0.7 MeV
Behavior of Ecm from PBV is similar to PAV.
Energy E from PBV is similar to PAV after the first RHF iteration (G1), but very different from the second RHF iteration (G2).

35. Center-of-mass Motion
Single-particle spectrum for PAV and PBV for 1st RHF iteration (G1), and the final RHF iteration (G4).
As PBV includes -Hcm in RHF equation, the sp energies are lower than that of PAV, especially those high lying states.
Different s.p. spectrum give different G-matrix.
PAV is a good approximation for PBV in RHF calculation, but not in RBHF calculation.
In RBHF calculation, PBV method should be used.

36. Self-Consistent Basis for RBHF
Total energy of RBHF calculation at each iteration step with different initial DWS basis
Self-consistency is very important for RBHF calculation.

37. Different Choices for Single-Particle Potential
Uncertainty in the definition of s.p. potential for particle states (a, b > kF),
We will investigate the following two situations:
Gap choice : U = 0.
Similar as continuous choice but with ε’ = energy of occupied states, e.g. επ1p1/2 or εν1s1/2 .
Results
Choice ε’ = επ1p1/2 will be used.

38. Different Choices for Single-Particle Potential
Energy spectrum and charge density distribution of RBHF calculation for 16O with different s.p. potentials for particle states.
There is a “gap” between particle states and hole states with gap choice.
Choice with ε’ below Fermi surface give similar results.
Central density given by gap choice is smaller than the other choices.

39. RBHF for 16O
Total energies and charge radii of 16O obtained with RBHF, RBHF with EDA, and BHF.
For EDA and BHF: H. Müther, R. Brockmann, and R. Machleidt, PRC 42, 1981 (1990).
RBHF improves the description over EDA or BHF.
Binding energy and charge radius given by Bonn potentials are smaller than experimental data.

40. RBHF for 16O
Single-particle spectrums for Bonn A, B, C with RBHF, in comparison with experimental data L. Coraggio, et al., PRC 68, (2003).
Bonn A gives the best description within RBHF framework.
The gap at Fermi surface may be due to lack of higher order configurations
E. Litvinova and P. Ring, PRC 73, (2006).

41. RBHF for 16O
Charge density distributions for Bonn A, B, C with RBHF, in comparison with experiment H. De Vries, C. De Jager, and C. De Vries, At. Data Nucl. Data Tables 36, 495 (1987).
Central density distributions given by Bonn potentials are larger than data.
Density distribution gets more concentrated in the center: Bonn C  B  A.

42. Comparison for 16O
Energy, charge radius, matter radius, and π1p spin-orbit splitting in 16O with Bonn A/B/C in RBHF; Bonn A in nonrelativistic renormalized BHF H. Müther, R. Brockmann, and R. Machleidt, PRC 42, 1981 (1990); Vlow−k derived from Argonne v18 in BHF B. Hu, et al., arXiv: ; N3LO in no core shell model (NCSM) R. Roth, et al., PRL 107, (2011) and coupled-cluster theory (CC) G. Hagen, et al., PRC 80, (2009).
E (MeV)
rc (fm)
rm (fm)
ΔEπ1pls (MeV)
Exp.
-127.6
2.70
2.54
6.3
RBHF, Bonn A
-113.5
2.56
2.42
5.4
RBHF, Bonn B
-102.7
2.61
2.47
4.6
RBHF, Bonn C
-95.0
2.65
2.52
4.0
BHF, Bonn A
-105.0
2.29 -
7.5
BHF, Vlow−k
-134.2
1.92
13.0
NCSM, N3LO
-119.7(6)
CC, N3LO
-121.0
2.30
Relativistic effect is important to improve description.
The results of RBHF are similar to other state-of-the-art ab initio results.

43. Comparison for 4He
Energy, charge radius and proton radius of 4He. Results of Bonn A in RBHF with PBV and PAV, comparing with CD-Bonn in Faddeev-Yakubovsky (FY) equations A. Nogga, H. Kamada, and W. Glöckle, PRL 85, 944 (2000 N4LO in FY S. Binder, et al., PRC 93, (2016); N2LO in nuclear lattice effective field theory (NLEFT) T. Lähde, et al., PLB 732, 110 (2014); N3LO in NCSM P. Navrátil, Few Body Syst. 41, 117 (2007); Vlow−k derived from Argonne v18 in BHF B. Hu, et al., arXiv:
E (MeV)
rp (fm)
Exp.
-28.3
1.46
RBHF (PBV)
-35.0
1.64
RBHF (PAV)
-26.3
1.73
FY, CD-Bonn -
FY, N4LO
-24.3
1.55
NLEFT, N2LO
-25.6
NCSM, N3LO
-25.3
BHF, AV18
-25.9
Treatment of center-of-mass motion need to be careful for light system.
Radii given by ab initio calculations are generally larger than data.

44. RBHF for 40Ca and 48Ca
Energies, charge radii, matter radius, and π1d spin-orbit splittings of 40Ca and 48Ca. Results of Bonn A in RBHF, comparing with N3LO in CC G. Hagen, et al., PRC 82, (2010); Vlow−k derived from Argonne v18 in non-relativistic BHF B. Hu, et al., arXiv: , NCSM R. Roth, et al., PRL 99, (2007); and CC G. Hagen, et al., PRC 76, (2007).
40Ca
48Ca
E (MeV)
rc (fm)
rm (fm)
ΔEπ1dls (MeV)
Exp.
-342.1
3.48 -
6.6 ± 2.5
-416.1
4.7
RBHF, Bonn A
-290.8
3.23
3.11
5.9
-349.6
3.27
2.7
CC, N3LO
-345.2
-396.5
BHF, Vlow−k
-552.1
2.20
24.9
NCSM, Vlow−k
-461.8
2.27
CC, Vlow−k
-502.9
Storage: GB
CPU time: 1720 h
Storage: GB
CPU time: 4900 h
For RBHF
Results for 40Ca and 48Ca given by RBHF are similar as for 16O.
Radius given by ab initio calculations are generally larger than experiment.

45. RBHF for 40Ca and 48Ca
Charge density distribution of 40Ca and 48Ca calculated by RBHF theory with Bonn A potential, in comparison with experimental data H. De Vries, C. De Jager, and C. De Vries, At. Data Nucl. Data Tables 36, 495 (1987).
Central density given by RBHF theory is higher than experimental value, while the relation between 40Ca and 48Ca is similar as experiment.

46. RBHF for Neutron Skin of 48Ca
Neutron skin of 48Ca by RBHF theory with Bonn A potential, in comparison with ab initio CC calculation G. Hagen, et al., Nat. Phys. 12, 186 (2015)., other theoretical analysis (quasielastic charge-exchange reactions P. Danielewicz, P. Singh, and J. Lee, NPA 958, 147 (2017), Coulomb energies J. Bonnard, S. Lenzi, and A. Zuker, PRL 116, (2016)), and experiment J. Birkhan, et al., arXiv: , X. Roca-Maza, et al., PRC 92, (2015).
Neutron skin of 48Ca given by RBHF with Bonn A potential agrees with other ab initio calculation and experiment.

47. RBHF for 40Ca and 48Ca
Single-particle spectrum and localized 1s1/2 single-particle potential of 40Ca (left) and 48Ca (right) in RBHF with Bonn A interaction, in comparison with experimental data L. Coraggio, et al., PRC 68, (2003), A. Bouyssy, et al. PRC 36, 380 (1987).
RBHF theory is very promising to describe the heavy nuclei.

48. Summary and Perspectives
Full relativistic Brueckner-Hartree-Fock equations have been solved self-consistently for the first time for finite nuclei.
16O is taken as an example to show the convergence
16O, 40Ca and 48Ca have been studied with Bonn A interaction. The resulting binding energies and charge radii are reasonably good. The spin-orbit splittings, and the neutron skin of 48Ca are well reproduced.

49. Thank you for your attention!
Summary and Perspectives
Future Work
To parallelize the RBHF code and calculate on supercomputer for heavy nuclei.
To learn from the G-matrix and provide information to nuclear energy density functional and other studies.
To investigate with relativistic LQCD force or relativistic chiral force Ren et al. arXiv:
To investigate the three-body force in RBHF framework.
To improve the convergence, e.g. by using V-low k or SRG technique.
To investigate the effect of higher orders in the hole-line expansion beyond RBHF.
...
Thank you for your attention!

50. Ground state properties
CDFT, implemented with self-consistency and taking into account various correlations by spontaneously broken symmetries, provide an excellent description for the ground- state properties including total energy and other physical observables as the expectation values of local one-body operators
Even for exotic nuclei with extreme neutron or proton numbers, where novel phenomena such as halos may appear.
Meng, Toki, Zhou, Zhang, Long, Geng, Prog. Part. Nucl. Phys. 57 (2006) 470
Meng and Zhou, J. Phys. G: Nucl. Part. Phys. 42 (2015)

51. the binding energies of the lowest mean-field states;
CDFT calculated binding energies by PC-PK1 with the data for 575 even–even nuclei:
the binding energies of the lowest mean-field states;
including the rotational correction energies;
the full dynamical correlation Energies.
Zhang, Niu, Li, Yao, Meng, Front. Phys. 9(2014) 529
PC-PK1
Zhao, Li, Yao, Meng, PRC 82, (2010)

52. See also: Afanasjev, Agbemava, Ray, Ring, PLB726(2013)680
Drip-lines in variant models
The number of bound nuclides with between 2 and 120 protons is around 7, J U N E | V O L | N AT U R E | 5 0 9
Figure: bound nuclei from Z=8 to Z=130 predicted by RCHB theory with PC-PK1. For 2227 nuclei with data, binding energy differences between data and calculated results are shown in different color. The nucleon drip-lines predicted TMA, HFB-21, WS3, FRDM , UNEDF and without pairing correlation are plotted for comparison.
See also: Afanasjev, Agbemava, Ray, Ring, PLB726(2013)680

53. CDFT in a static external field
CDFT in a static external field includes:
Constrained CDFT is a powerful tool to investigate the shape evolution, shape isomers, shape-coexistence, and fission landscapes.
Cranking CDFT obtained by transforming from the laboratory to the intrinsic frame is widely used to describe rotational spectra in near spherical, deformed, superdeformed, and triaxial nuclei.
Review on cranking CDFT:
Vretenar, Afanasjev, Lalazissis, Ring, Physics Reports 409 (2005)
Meng, Peng, Zhang, Zhao, Front. Phys. 8 (2013) 55-79

54. Multidimentionally constrained CDFT
β2, β3, …: Geng, Meng, Toki, 2007, Chinese Phys. Lett
β2, γ, …: Meng, Peng, Zhang, Zhou, 2006, PRC
MDC-CDFT: all blm with even m included
Triaxial & octupole shapes both crucial around the outer barrier
Multidimensionally constrained covariant density functional theories (MDC-CDFTs) have been developed. In these models, all shape degrees of freedom \beta_{\lambda\mu} with even \mu are allowed, e.g., \beta_{20}, \beta_{22}, \beta_{30}, \beta_{32}, and so on.
MDC-CDFTs have been used to investigate the potential energy surfaces and fission barriers of actinide nuclei, including possible triple-humped structure in barriers, and the non-axial (Y32) correlations in N=150 isotones. For example, in the study of fission barriers, it was found that the triaxial deformation lowers not only the first, but also the second fission barriers considerably.
Further information:
MDC-CDFTs: the first CDFT with both non-axial and reflection asymmetric shapes.
For the first time, the influence of triaxiality on the outer barrier height was revealed: The triaxial deformation lowers the outer barrier height by 0.5 to 1 MeV for even-even actinide nuclei, amounting for 10 to 20% of the total height.
Results in this slide: PC-PK1; BCS with delta force
Figure on the left [Fig. 1 in Lu, Zhao, Zhou, PRC 85, (2012)]: Potential energy curves of 240Pu with various self-consistent symmetries imposed. The solid black curve represents the full calculation, i.e., with all shape degrees of freedom \beta_{\lambda\mu} with even \mu allowed, the red dashed curve that with axial symmetry (AS) imposed, the green dotted curve that with reflection symmetry (RS) imposed, the violet dot-dashed line that with both symmetries (AS & RS) imposed. The empirical inner (outer) barrier height B_\mathrm{emp} is denoted by the grey square (circle). The energy is normalized with respect to the binding energy of the ground state.
Figure on the right [Fig. 3 in Lu, Zhao, Zhou, PRC 85, (2012)]: Sections of the three-dimensional PES of 240Pu in the (\beta_{22},\beta_{30}) plane calculated at \beta_{20}= 0.3 (around the ground state), 0.6 (around the first saddle point), 0.9 (around the fission isomer), 1.3 (around the second saddle point) and 1.6 (beyond the outer barrier), respectively. The energy is normalized with respect to the binding energy of the ground state. The contour interval is 0.5 MeV. Local minima are denoted by crosses.
In MDC-CDFTs, the functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the non-linear or density-dependent couplings; pairing is treated either by BCS (MDC-RMF) or Bogoliubov (MDC-RHB) with delta force or separable finite-range force; axially deformed harmonic oscillator (ADHO) basis
Figure: Potential energy curve of 240Pu
Lu, Zhao, Zhou, PRC 85, (2012)
Zhao, Lu, Zhao, Zhou, PRC 86, (2012)
Lu, Zhao, Zhao, Zhou, PRC 89, (2014)
Zhao, Lu, Vretenar, Zhao, Zhou, arXiv: (2014)
Figure: 3D PES of 240Pu
Abusara, Afanasjev, and Ring, PRC 85, (2012)
courtesy of B.N. LU

55. Time-dependent CDFT
For excited states, a proper method is time-dependent CDFT.
In analogy to the Hohenberg-Kohn theorem, there exists the Runge-Gross theorem.
Runge-Gross theorem provides an exact mapping of the full time-dependent many-body problem onto a time- dependent single-particle problem.
The corresponding single-particle field is not only time- dependent, but also depends on the single-particle density with its full time dependence, i.e., it includes memory effects.

56. Time-dependent CDFT
For vibrations with small amplitude, it can be expanded in the vicinity of the ground state and a connection to the static energy density functional can be found.
In the adiabatic approximation, i.e. by neglecting the memory effects and the energy dependence in Fourier space, one ends up with the RPA with a residual interaction derived as the second derivative of the static energy functional with the density.
RPA provides a successful description of the mean energies of giant resonances in nuclei, but not able to reproduce the decay width of these excitations.
For the decay width and the fragmentation of single-particle states, one has to go beyond mean field and to consider the energy dependence of the self-energy. This can be done within the particle-vibrational coupling (PVC) model.
Paar, Vretenar, Khan, Colo, Rep. Prog. Phys. 70, 691 (2007).

57. RHB+QRPA: Nuclear β+/EC-decay half-lives
RHB+QRPA: well reproduces the experimental half-lives for neutron-deficient Ar, Ca, Ti, Fe, Ni, and Zn isotopes by a universal T=0 pairing strength.
FRDM+QRPA: systematically overestimates the nuclear half-lives  the pp residual interactions in the T=0 channel are not considered.
Nuclear β+/EC-decay half-lives calculated in RHB+QRPA model with the PC-PK1 parameter set.
Niu et al., PRC 87, (R) (2013)

58. RHB+QRPA/PC-PK1: Nuclear β-decay half-lives
Z. Y. Wang, Z. M. Niu, Y. F. Niu, J. Y. Guo arXiv:
Nuclear $β$-decay half-lives in the relativistic point-coupling model

59. Symmetry restoration and fluctuations
Many-body energy takes the form of a functional of all transition density matrices between the various Slater determinants of different deformation and orientation. Mixing of these different configurations allows the restoration of symmetries and to take into account fluctuations around the mean-field equilibrium solution.
In full space and with deformation degrees of freedom, such calculations require considerable numerical efforts, at the limit of present computer capabilities.
A considerable simplification is by using the constraint calculations for the derivation of a collective Hamiltonian in these degrees of freedom.

60. Yao, Hagino, Li, Meng, Ring, Phys. Rev. C 89, 054306 (2014)
Nuclear low-lying states: CDFT+GCM+PN3DAMP
7D GCM: two deformation parameters + projection 3DAM and 2PN
The low-energy spectrum in 76Kr are well reproduced after including triaxiality in the full microscopic GCM+ PN3DAMP calculation based on the CDFT using PC-PK1.
This study answers the important question of dynamic correlations and triaxiality in shape-coexistence nucleus 76Kr and provides the first benchmark for the EDF based collective Hamiltonian method.
Yao, Hagino, Li, Meng, Ring, Phys. Rev. C 89, (2014)
Benchmark for the collective Hamiltonian in five dimensions

61. Simultaneous shape phase transition
5DCH Calculations based on CDFT PC-PK1 indicate a simultaneous quantum shape phase transition from spherical to prolate shapes, and from reflection symmetric to octupole shapes.
triaxial quadrupole energy surfaces
axially-symmetric quadrupole–
octupole energy surfaces
The evolution of quadrupole and octupole shapes in Th isotopes is studied in the framework of CDFT. Constrained energy maps and observables calculated with microscopic collective Hamiltonians indicate the occurrence of a simultaneous quantum shape phase transition between spherical and quadrupole-deformed prolate shapes, and between non-octupole and octupole-deformed shapes, as functions of the neutron number. The nucleus 224Th is closest to the critical point of a double phase transition.
probability density distributions for the ground states
Li, Song, Yao, Vretenar, and Meng, Phys. Lett. B 726 (2013) 866
2/16/2018

62. Relativistic description of nuclear matrix elements in neutrinoless double-β  decay
Phys. Rev. C 90, – Published 10 November 2014
Song, Yao, Ring, and Meng
Phys. Rev. C 91, (2015)
Yao, Song, Hagino, Ring, and Meng