Two approximate approaches for solving the large-scale shell model problem

1. Two approximate approaches for solving the large-scale shell model problem Sevdalina S. Dimitrova Institute for Nuclear Research & Nuclear Energy Bulgarian Academy of Sciences

2. Collaborators • ISA Nicola Lo Iudice University of Naples, Italy Antonio Porrino University of Naples, Italy Francesco Andreozzi University of Naples, Italy Davide Bianco University of Naples, Italy • DMRG Jorge Dukelsky Instituto de Estructura de la Materia, Madrid,Spain Stuart Pittel Bartol Research Institute, University of Delaware, USA Mario Stoitsov Institute for Nuclear Research & Nuclear Energy, Sofia

3. Contains • Large - scale shell model • Density matrix renormalization group method • Importance sampling algorithm • Calculations for 48Cr in the fp-shell

4. Large-scale shell model • Hamiltonian:

5. Configuration space • fp -shell

6. DMRG • Goal of the project: • Develop the Density Matrix Renormalization Group (DMRG) method for use in nuclear structure; • Background: • DMRG method introduced by Steven White in 1992 as an improvement of Ken Wilson’s Renormalization Group; • Used extensively in condensed matter physics and quantum chemistry; • DMRG principle: • Systematically take into account the physics ofall single-particle levels: • Still the ordering of the single-particle levels plays a crucial role for the convergence of the calculations;

7. DMRG • Q: How to construct optimal approximation to the ground state wave function when we only retain certain number of particle and hole states? • A: • Choose the states that maximize the overlap between the truncated state and the exact ground state. • Q: How to do this? • A: • Diagonalize the Hamiltonian • Define the reduced density matrices for particles and holes

8. DMRG • Diagonalize these matrices: P,Hrepresent the probability of finding a particular -state in the full ground state wave function of the system; Optimal truncationcorresponds to retaining a fixed number of eigenvectors that have largest probability of being in ground state, i.e., have largest eigenvalues; • Bottom line:DMRG is a method for systematically building in correlations from all single-particle levels in problem. As long as convergence is sufficiently rapid as a function of number of states kept, it should give an accurate description of the ground state of the system, without us ever having to diagonalize enormous Hamiltonian matrices;

9. DMRG • Subtleties: • Must calculate matrix elements of all relevant operators at each step of the procedure. • This makes it possible to set up an iterative procedure whereby each level can be added straightforwardly. Must of course rotate set of stored matrix elements to optimal (truncated) basis at each iteration. • Procedure as described guarantees optimization of ground state. To get optimal description of many states, we may need to construct mixed density matrices, namely density matrices that simultaneously include info on several states of the system.

10. Sph HF 24Mg in m - scheme • sd-shell • 4 valent protons • 4 valent neutrons • USD interaction

11. Importance Sampling F. Andreozzi, A. Porrino and N. Lo Iudice, “A simple iterative algorithm for generating selected eigenspaces of large matrices” J. Phys. A: 35 (2002) L61–L66 • an iterative algorithm for determining a selected set of eigenvectors of a large matrix, robust and yielding always to ghost-free stable solutions; • algorithm with an importance sampling for reducing the sizes of the matrix, in full control of the accuracy of the eigensolutions;

12. µ ¶ ^ ¸ b f f g ( g f h j ) j i g j i j i f j i j i j i g H H H H N i j i j i j 1 2 1 2 ^ j j 1 ¡ = = = = i j i j h j j i h b Á f H N j j 3 ; ; ; ; ; : : : ; w e r e o r = = j j 1 ¡ b ; : : : ; H j j j ( ) ( ) ( ) N N 2 2 ( ) ( ) 1 1 P j i j j i i j j i i j i ¸ Á ¸ Ã Á E i 1 2 + ´ ´ c c c = = N N 2 2 1 2 i i 1 ; ; ; = Importance Sampling • the iterative dialgonalization algorithm: • zero approximation loop: • diagonalize the two-dimensional matrix • select the lowest eigenvalue and the corresponding eigenvector: • diagonalize the two-dimensional matrix • select the lowest eigenvalue and the corresponding eigenvector ………. eigenvalue problem of general form • approximate eigenvalue and eigenvector

13. Ã ! ~ f g ( ) H i j 1 b ¤ m = i j ~ j ; ; v f g h b b b b H w e r e = = j j j j 1 2 ~ v T ; ; ; : : : ; b H j j j P 0 ( j 0 ) j ¸ ¸ ¸ i 1 ¡ > v ² = i i i i 1 ; ; v = ; Importance Sampling • the importance sampling algorithm • start with m basis (m >v) vectors and diagonalize the m-dimensional principal submatrix • for j = v+1, …N diagonalize the v+1-dimensional matrix : • select the lowest eigenvalues and accept the new state only if • When the truncated configuration space is determined, apply the iterative diagonalization algorithm

14. Importance Sampling • The algorithm has been shown to be completely equivalent to the method of optimal relaxation of I. Shavitt and has therefore a variational foundation; • It can be proven that the approximate solution of the eigenvalue problem converges to the exact one; • A generalization to calculate several eigenvalues and eigenvectors is straightforward

15. 48Cr in the fp shell: j-scheme

16. · ¸ N E E A + ¡ e x p = 0 c 48Cr in the fp shell: m-scheme

17. 48Cr in the fp shell: m-scheme

18. 48Cr in the fp shell: m-scheme

19. J ¼ " " " " D M R G H F D M R G I S A + t e x a c + 0 3 2 2 4 9 3 2 8 4 0 3 2 9 1 3 3 2 9 5 3 - - - - . . . . + 2 3 1 6 5 0 3 2 0 1 6 3 2 0 9 8 3 2 1 4 8 - - - - . . . . + 4 3 1 1 4 9 3 1 6 6 8 3 1 1 1 1 3 1 1 2 8 - - - - . . . . 48Cr: m-scheme in the fp shell

20. Conclusions: • The first calculations within the ISA in the m-scheme prove the applicability of the method to large-scale shell-model problems. • The DMRG is also a practical approach which needs more tuning. • At it is, the ISA code requires a lot of disk space for considering 56Ni for example. • At it is, the DMRG code requires large RAM memory to describe heavier nuclei. • Both methods are applicable for odd mass nuclei as well.