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

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

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

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, 034323 (2003); Zhou, Meng, Ring, and Zhao, PRC 82, 011301 (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; …

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).

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, 057301 (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.

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 16 O, 20 Ne and 208 Pb excellently. Furthermore, the rod shapes of 12 C , 16 O , 24 Mg are investigated.

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

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

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, 057301 (2010);

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, 057301 (2010);

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

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).

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),

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.

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

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 .

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!

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. 32 24 (1979)

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 2.84 (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)

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),

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

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 .

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

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

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 .

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.

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

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.

Deformed single particle levels

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

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.

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] 127.285 127.286 127.230 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] 1637.941 1637.915 1637.780 Charge Radius 𝒓 𝒄 [fm] 5.518 5.517 𝒓 𝒏 −𝒓 𝒑 [fm] 0.257

Functional PC-PK1: deformed nuclei Nucleus: 𝟐𝟎 Ne RMF3D (n=24, d=0.8fm) DIZ (nf=18) Binding Energy [MeV] 155.552 155.509 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.

Functional DD-ME2: spherical nuclei Nucleus: 𝟏𝟔 O RMF3D (n=24, d=0.8 fm) Lalazissis et al PRC71, 024312 (2005) Absolute deviation Binding Energy [MeV] 127.809 127.801 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, 024312 (2005) Absolute de viation Binding Energy [MeV] 1638.483 1638.426 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.

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

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, 022501 (2015) ; Yao, Itagaki, and Meng, PRC 90, 054307(2014); Iwata, Ichikawa, Itagaki, Maruhn, and Otsuka, PRC 92, 011303(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.

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

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

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 = 16.48 MeV The convergence of 3D lattice calculation is very good.

(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 = 35.55 MeV 𝟐𝟒 𝐌𝐠 𝐸 excited = 60.35 MeV

(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.

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

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.

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

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),