1. Supervisor: Prof. Jie Meng School of Physics, Peking University
Covariant Density Functional Theory in 3D lattice based on inverse Hamiltonian method and spectral method
Zheng-xue Ren (任政学)
Supervisor: Prof. Jie Meng
School of Physics, Peking University
Collaborator: Prof. Shuang-quan Zhang

2. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

3. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

4. Introduction To get a better convergence description of the nuclei
far from the β-stability line;
with exotic excitation modes;
one should investigate them in coordinate space or coordinate-equivalent space.
In CDFT framework, the existing methods
Runge-Kutta / Shooting method Meng, NPA 635, 3 (1998)
Dirac Woods-Saxon basis Zhou, Meng, and Ring, PRC 68, (2003);
Zhou, Meng, Ring, and Zhao, PRC 82, (2010)
However, it’s difficult to extend these methods to 3D case.
Imaginary time method (ITM) is a good choice for the calculation in coordinate space. Davies, Flocard, Krieger, and Weiss, NPA 342, 111 (1980).
with exotic shapes; …

5. Introduction
ITM is a iterative method for self-consistent mean field problem, and it has been applied to nonrelativistic mean field calculations in 3D coordinate space successfully. Bonche, Flocard, and Heenen, Comput. Phys. Comm. 171, 49 (2005); Maruhn, Reinhard, Stevenson, and Umar, Comput. Phys. Comm. 185, 2195 (2014).
However, there are two challenges when one applies ITM to CDFT naively,
Variational collapse;
Fermion doubling (spurious solutions).
To avoid these problems, Zhang, Liang and Meng applied the ITM to the Schr o dinger-equivalent form of the Dirac equation, and developed a spherical RMF code based on this method. Zhang, Liang, and Meng, CPC 33, 113 (2009); Zhang, Liang, and Meng, IJMPE 19, 55 (2010).

6. Introduction
To avoid variational collapse, Hagino and Tanimura proposed inverse Hamiltonian method (IHM); to avoid Fermion doubling problem, Tanimura Hagino and Liang introduced high-order Wilson terms. Based on these two methods, they performed 3D lattice calculations for CDFT. Hagino and Tanimura, PRC 82, (2010); Tanimura, Hagino, and Liang, PTEP 2015, 073D01 (2015).
However, there are some drawbacks when employing high-order Wilson terms,
Correcting energies and wave functions.
Introducing artificial violation of the rotational symmetry.

7. Introduction In this work:
To avoid variational collapse problem, we employ inverse Hamiltonian method.
To avoid Fermion doubling problem, we perform the spatial derivatives of Dirac equation in momentum space with the help of discrete Fourier transform (spectral method). S. J. Shen, T. Tang, and L. L. Wang, Spectral methods : algorithms, analysis and applications (Springer, 2011).
Combining these two methods, we propose a new method to solve Dirac equation in 3D lattice space.
Then we implant this new method on CDFT, and reproduce the properties of O, Ne and Pb excellently. Furthermore, the rod shapes of 12 C , O , 24 Mg are investigated.

8. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

9. Imaginary time method
If Hamiltonian ℎ is independent on time, the evolution of |𝜓 0 ⟩ reads
Basic idea of ITM is 𝑡→−i𝜏, and now the evolution of |𝜓 0 ⟩ becomes
where
Obviously, with𝜏→∞, | 𝜓 (𝜏)⟩will converge to the ground state of Hamiltonian ℎ. Davies, Flocard, Krieger, and Weiss, NPA 342, 111 (1980).
In practice, the imaginary time 𝜏 is cut into steps by 𝛥𝜏, and the evolution is carried out iteratively. Then, the evolution operator is expanded to the linear order of 𝛥𝜏

10. Inverse Hamiltonian method
Because of the Dirac sea, a direct extension of ITM to Dirac equation will meet so-called variational collapse problem. Zhang, Liang, and Meng, CPC 33, 113 (2009); Zhang, Liang, and Meng, IJMPE 19, 55 (2010).
To avoid variational collapse in Ritz variational principle – maximizing ⟨1/ℎ⟩ instead of minimize ⟨ℎ⟩ Hill and Krauthauser, PRL 72, 151(1994)
Hagino and Tanimura combined this idea with ITM and proposed IHM. The evolution of wave functions reads 𝜓 𝜏 = e 𝜏 ℎ−𝑊 𝜓 0 , who will converge to the ground state in Fermi sea as 𝜏→∞ with a suitable 𝑊. Hagino and Tanimura, PRC 82, (2010);

11. Inverse Hamiltonian method
Because of the Dirac sea, a direct extension of ITM to Dirac equation will meet so-called variational collapse problem. Zhang, Liang, and Meng, CPC 33, 113 (2009); Zhang, Liang, and Meng, IJMPE 19, 55 (2010).
To avoid variational collapse in Ritz variational principle – maximizing ⟨1/ℎ⟩ instead of minimizing ⟨ℎ⟩ Hill and Krauthauser, PRL 72, 151(1994)
Hagino and Tanimura combined this idea with ITM and proposed IHM. The evolution of wave functions reads 𝜓 𝜏 = e 𝜏 ℎ−𝑊 𝜓 0 , who will converge to the ground state in Fermi sea as 𝜏→∞ with a suitable 𝑊. Hagino and Tanimura, PRC 82, (2010);

12. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

13. Fermion doubling problem
Solving Dirac equation in lattice space one may get spurious states with high momentum but low energy. Wilson1977Quarks and strings on a lattice
1D free Dirac equation:
its solutions have the following form
Taking finite difference method,
one will get
Dispersion relation 𝜀 2 = 𝑘 2 + 𝑚 2 becomes
Taken from Tanimura, Hagino, and Liang, PTEP 2015, 073D01 (2015).

14. Spectral method In momentum space, 1D free Dirac equation
It gives exact dispersion relation: 𝜀 2 = 𝑘 2 + 𝑚 2 , i.e. no Fermion doubling.
In lattice space, spatial derivatives can be performed in momentum space with the help of discrete Fourier transform. In numerical physics, this method is so-called spectral method.
Assuming 𝑓(𝑥 𝜈 ) in momentum space is 𝑓 (𝑘 𝑛 ), 𝑚-th derivative of 𝑓(𝑥 𝜈 )
𝑓 (𝑚) (𝑥 𝜈 ) can be get from following procedure (FT: Fourier transform),

15. Grids in coordinate and momentum space
The grids in coordinate space 𝒓 𝑥 ν 𝑥 , 𝑦 ν 𝑦 , 𝑧 ν 𝑧
The grids in momentum space
y- and z- directions are similar to x-direction.

16. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

17. Self-consistent calculation
The procedure to perform the self-consistent calculation based inverse Hamiltonian method are listed. Tanimura, Hagino, and Liang, PTEP 2015, 073D01 (2015).
Step 1 : Prepare a set of initial single-particle wave functions { 𝜓 𝑘 (0) } with 𝑘=1, 2, …,𝐴.
Step 2 : Calculate the densities 𝜌 (𝑛) and currents 𝒋 (𝑛) from { 𝜓 𝑘 (𝑛) }.
Step 3 : Construct the single particle Hamiltonian ℎ (𝑛) .
Step 4 : Generate the set wave functions { 𝜑 𝑘 (𝑛+1) }, via the evolution
Step 5 : Orthonormalize { 𝜑 𝑘 (𝑛+1) } to obtain the { 𝜓 𝑘 (𝑛+1) }, if the dispersions of energy, , for interesting levels, are not smaller enough, go to Step 2 .

18. The Coulomb field solver
For the Possion equation of Coulomb field,
its solution have the form
Using the property of Laplace operator,
the integration can be reduced as
If one calculates it directly, a 3D integration is required for each point 𝒓. For example, 30×30×30 grids means 27,000 integrations.
Too expensive!

19. The Coulomb field solver
Coulomb field can also be obtained based on discrete Fourier transform.
Note that the integration for Coulomb field 𝐴 0 (𝒓) is a convolution
A procedure to obtain the Coulomb field 𝐴 0 𝒓 is provided by Eastwood and Brownrigg. Eastwood, and Brownrigg, J. Comp. Phys (1979)

20. The Coulomb field solver
Step 1. Extend lattice space from nx×ny×nz to 2nx×2ny×2nz.
Step 2. Get extended density 𝜌 2 (𝒓)
Step 3. Calculate 1/𝒓 in 2nx×2ny×2nz lattice space, where the value on origin is replaced by (dxdydz) 1/3 , the obtained function denoted as 𝑞(𝒓).
Step 4. Compute the discrete Fourier transform 𝜌 2 (𝒌) of 𝜌 2 (𝒓), and 𝑞 𝒌 of 𝑞(𝒓).
Step 5. Get the extended Coulomb field in momentum space
Step 6. Take a inverse discrete Fourier transform on 𝐴 2 0 (𝒓), then get 𝐴 0 (𝒓)

21. The meson field solver In meson exchange model, the equation of meson,
is solved in momentum space.
In momentum space, the mesons’ equation reads
therefore 𝜙 (𝒌) can be obtained by 𝑆 𝜙 (𝒌),
In summary, 𝜙(𝒓) can be get from following procedure (FT: Fourier transform),

22. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

23. Numerical details Imaginary time step size: 𝛥𝜏=100 MeV.
The inverse of Hamiltonian is calculated by conjugate residual method. S. Yousef, Iterative methods for sparse linear systems, (Siam, 2003)
The energy shift 𝑊 for different level is taken as
where ∆ 𝑊 1 =6 MeV and for 𝑖>1,
The Dirac Hamiltonian is diagonalized within the space of the evolution wave functions every 10 iterations, and the eigenfunctions thus obtained are taken as then initial wave functions for future iteration.
The convergence is determined by the energy dispersions for interesting states are smaller than 10 −4 MeV .

24. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

25. Woods-Saxon type potential
Potential of Dirac equation
with
The box sizes and step sizes for three direction are identical, denoted as 𝐿 and 𝑑 respectively. In calculations, 𝐿=23 fm and 𝑑=1fm if not specified.
𝑉 𝟎 [MeV]
𝑅 𝟎 [fm]
𝑎 [𝐟𝐦] λ
𝑅 𝑙𝑠 [𝐟𝐦]
𝑎 𝑙𝑠 [𝐟𝐦]
−65.796
4.482
0.615
11.118
4.159
0.648

26. Spherical single particle levels
For clarify, only one energy level for each degenerate orbital is shown.
After 39-th iteration, all the energy dispersions of bound sates are smaller than 10 −4 MeV .

27. Energy accuracy Fig.(a) and Fig.(b): 𝐿 𝑎 ≈𝐿 𝑏 ≈23 fm
𝑑 𝑎 =1.0 fm, 𝑑 𝑏 =0.8 fm
Except weak bound 2 p 3/2 and 2 p 1/2 orbitals, the deviations are smaller than 10 −4 MeV.
Fig.(b) and Fig.(c):
𝐿 𝑏 =23.2 fm, 𝐿 𝑐 =31.2 fm
𝑑 𝑏 = 𝑑 𝑐 =0.8 fm
the deviations of 2 p 3/2 and 2 p 1/2 orbitals drop obviously, whereas the others are almost unchanged.

28. Density distribution
The distribution of total density of six states of 1 d 5/2 orbit is spherical.

29. Radial density distribution
In 3D lattice calculations, radial (vector) densities are obtained by following equation
The radial densities obtained by shooting method and 3D lattice calculations are almost identical.

30. Deformed single particle levels

31. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

32. Information about CDFT calculation
The relativistic nonlinear point-coupling model and relativistic density- dependent meson-exchange model are employed.
In the calculation, the mesh numbers and step sizes are denoted as (n, d) for the case that mesh numbers and step sizes of three direction are identical, and (nx, ny, nz), (dx, dy, dz) for then general case.
The comparisons between 3D lattice calculation (RMF3D) and spherical code calculation (DIR), axial deformed code calculation (DIZ), are make.
All the calculations take microscopic center-of-mass correction.

33. Functional PC-PK1: spherical nuclei
Nucleus: 𝟏𝟔 O
RMF3D
(n=24, d= 0.8 fm)
DIR
(R=20 fm, dr=0.1fm)
DIZ
(Nf=18)
Binding Energy [MeV]
Charge Radius 𝒓 𝒄 [fm]
2.767
2.768
𝒓 𝒏 −𝒓 𝒑 [fm]
−0.023
−0.024
Nucleus: 𝟐𝟎𝟖 Pb
RMF3D
(n=30, d= 1 fm)
DIR
(R=20 fm, dr=0.1fm)
DIZ
(Nf=18)
Binding Energy [MeV]
Charge Radius 𝒓 𝒄 [fm]
5.518
5.517
𝒓 𝒏 −𝒓 𝒑 [fm]
0.257

34. Functional PC-PK1: deformed nuclei
Nucleus: 𝟐𝟎 Ne
RMF3D
(n=24, d=0.8fm)
DIZ
(nf=18)
Binding Energy [MeV]
Charge Radius 𝒓 𝒄 [fm]
3.007
3.006
𝒓 𝒏 −𝒓 𝒑 [fm]
−0.030
−0.029 𝜷
0.541
𝜷 𝟒𝟎
0.494
0.491
RMF3D gives a reliable results in the framework of relativistic point coupling model.

35. Functional DD-ME2: spherical nuclei
Nucleus: 𝟏𝟔 O
RMF3D
(n=24, d=0.8 fm)
Lalazissis et al PRC71, (2005)
Absolute deviation
Binding Energy [MeV]
0.008
Charge Radius 𝒓 𝒄 [fm]
2.729
2.727
0.002
𝒓 𝒏 −𝒓 𝒑 [fm]
-0.03
0.00
Nucleus: 𝟐𝟎𝟖 Pb
RMF3D
(n=30, d=1 fm)
Lalazissis et al PRC71, (2005)
Absolute de viation
Binding Energy [MeV]
0.057
Charge Radius 𝒓 𝒄 [fm]
5.519
5.518
0.001
𝒓 𝒏 −𝒓 𝒑 [fm]
0.19
0.00
RMF3D gives a reliable results in the framework of relativisitc meson exchange model.

36. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

37. Rod shape of 𝟏𝟐 𝐂 , 𝟏𝟔 𝐎 , 𝟐𝟒 𝐂
In the rod shapes of 𝑁=𝑍 nuclei such as 12 C , 16 O , 24 Mg , etc, the α-cluster structures have been discussed extensively with in density functional theory. Zhao, Itagaki, and Meng, PRL 115, (2015) ; Yao, Itagaki, and Meng, PRC 90, (2014); Iwata, Ichikawa, Itagaki, Maruhn, and Otsuka, PRC 92, (2015)
In CDFT framework, however, previous works on the rod shape are based on harmonic basis expansion with some symmetry restrictions. These symmetry restrictions will prevent the bending motion of the rod shapes.
In this work, the rod shapes of 12 C , 16 O , 24 Mg are investigated in 3D lattice calculation, where density functional DD-ME2 is used. The wave functions of 4 He are used to construct initial wave functions.

38. Rod shape of 𝟏𝟐 𝐂 : density in 𝒙𝒛 plane
Grids numbers and step sizes: (n, d)=( 40, 0.6 fm ).
The initial positions for three 4 He : (0, 0, -4 fm), (0, 0, 0), (0, 0, 4 fm).
Initial densities
Converged densities

39. Rod shape of 𝟏𝟐 𝐂 : density in xz plane
The initial positions for three 4 He :
(4sin 20° fm, 0, -4cos 20° fm), (0, 0, 0), (-4sin 20° fm, 0, 4cos 20° fm).
Initial densities
Converged densities

40. Convergence feature for the rod shape of 𝟏𝟐 𝐂
The convergence feature of various quantities with respect to the mesh numbers and step sizes is given in following table.
(nx, ny, nz)=
(dx, dy, dz)=
(20, 20, 30)
(0.8, 0.8, 0.8 )
(28, 28, 34)
(40, 40, 40)
(0.6, 0.6, 0.6)
Binding Energy [MeV]
71.269
71.272
Charge Radius 𝒓 𝒄 [fm]
3.368
3.367
𝒓 𝒏 −𝒓 𝒑 [fm]
-0.029 𝜷
2.481
2.480
𝜷 𝟒𝟎
4.787
4.788
𝑬 𝐞𝐱𝐜𝐢𝐭𝐞𝐝 = MeV
The convergence of 3D lattice calculation is very good.

41. (nx, ny, nz)=(30, 30, 40), (dx, dy, dz)=(0.8 fm, 0.8 fm, 0.8 fm)
Rod shape of 𝟏𝟔 𝐎 , 𝟐𝟒 𝐌𝐠 : density in 𝒙𝒛 plane
Grids numbers and step sizes:
(nx, ny, nz)=(30, 30, 40), (dx, dy, dz)=(0.8 fm, 0.8 fm, 0.8 fm)
𝟏𝟔 𝐎
𝐸 excited = MeV
𝟐𝟒 𝐌𝐠
𝐸 excited = MeV

42. (4sin θ fm, 0, -4cos θ fm), (0, 0, 0), (4sin θ fm, 0, 4cos θ fm)
The stability for the rod shape
In the calculation, if the initial positions of 4 He are not along a straight line, one is difficult to get the converged rod shape.
For example, in the calculation of 𝟏𝟐 𝐂 , the initial positions for three 4 He are taken as:
(4sin θ fm, 0, -4cos θ fm), (0, 0, 0), (4sin θ fm, 0, 4cos θ fm)
we can not get the rod shape solution even θ= 0.1°.
So the rod shape of 𝟏𝟐 𝐂 is highly unstable in this case (non-rotating case).
Next step we will investigate the stability of the rod shape in the rotating case, and seek the other possible exotic shapes.

43. Theoretical framework
Introduction
Theoretical framework
Inverse Hamiltonian method
Spectral method
Self-consistent calculation
Numerical details
Accuracy test
Dirac equations with a given potential
CDFT in 3D lattice
Rod shape of 1𝟐 C, 16 O, 24 Mg
Summary and perspectives

44. Summary and perspectives
In this work, Dirac equation is solved in 3D lattice by ITM. The variational collapse is avoided by invers Hamiltonian method; Fermion doubling problem is avoided by spectral method, i.e. performing spatial derivatives in momentum space by discrete Fourier transform.
The new method is used to solve Dirac equations with spherical, quadrapole, octupole potential efficiently.
Then, we realize 3D lattice calculations for CDFT, with which we investigate the rode shapes of 𝟏𝟐 𝐂 , 𝟏𝟔 𝐎 and 𝟐𝟒 𝐌𝐠 .
Perspectives
In the future, we will do following works:
3D Cranking CDFT;
Time-dependent CDFT.

45. Thank you !

46. Conjugate Residual Algorithm
Conjugate Residual Algorithm for linear equation 𝑨𝒙=𝒃:

47. Method to translate function
The translation of function can be realized by discrete Fourier transform.
For a function 𝑓(𝒓), its translation with R denotes as 𝑔(𝒓), which means 𝑔 𝒓 =𝑓(𝒓−𝑹) .
Take discrete Fourier transform on 𝑓 𝒓 ,
therefore we have discrete Fourier transform of 𝑔 𝒓
In summary, 𝜙(𝒓) can be get from following procedure (FT: Fourier transform),