1. Sevdalina S. Dimitrova Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria The Density Matrix Renormalization Group Method applied to Nuclear Shell Model Problems

2. Collaborators Jorge Dukelsky Instituto de Estructura de la Materia, Madrid Stuart Pittel Bartol Research Institute, University of Delaware, USA Mario Stoitsov Institute for Nuclear Research & Nuclear Energy, Sofia

3. Contains Introduction –Wilson’s Renormalization Group Method –Density Matrix Renormalization Group Method p-h DMRG basics Application to nuclear shell model problems Outlook

4. Wilson’s Renormalization Group (1974) The goal: to solve the Kondo problem (describes the antiferromagnetic interaction of the conduction electrons with a single localized impurity) after mapping it onto a 1D lattice in energy space. The assumption: low-energy states most important for law-energy behavior of large quantum systems

5. Wilson’s Renormalization Group (1974) The idea: numerically integrate out the irrelevant degrees of freedom The algorithm: → isolate finite subspace of the full configuration space → diagonalize numerically → keep m lowest energy eigenstates → add a site → iterate

6. Sampling the configuration space m sss Infinite procedure “the onion picture” superblock environment the size of the superblock stays the same while the environment shrinks

7. From WRG to DMRG The WRG was the first numerical implementation of the RG to a non-perturbative problem like the Kondo model, for which it had enormous success. WRG cannot be applied to other lattice problems. For 1D Hubbard models it begins to deviate significantly from the exact results. The problem resides in the fact that the truncation strategy is based solely on energy arguments. The solution to this problem was proposed by White who introducted the DMRG: PRL 69 (1992) 2863 and PR B 48 (1993) 10345.

8. From 1D lattices to finite Fermi systems S. White introduced the DMRG to treat 1D lattice models with high accuracy. PRL 69 (1992) 2863 and PR B 48 (1993) 10345. S. White and D. Husse studied S=1 Heisenberg chain giving the GS energy with 12 significant figures. PR B 48 (1993) 3844. T. Xiang proposed the k-DMRG for electrons in 2D lattices. PR B 53 (1996) R10445. S. White and R. L. Martin used the k-DMRG for quantum chemical calculation. J. Chem. Phys. 110 (1999) 4127. Since then applications in Quantum Chemistry, small metallic grain, nuclei, quantum Hall systems, etc… review article: U. Schollwöck, Rev. Mod. Phys. 77(2005)259

9. The particle-hole DMRG Introduced by J. Dukelsky and G.Sierra to study systems of utrasmall superconducting grains PRL 83 (1999) 172 and PRB 61 (2000) 12302 Motivation: BCS breaks particle number. PBCS improves the superconducting state. Fluctuation dominated phase? Level ordering: In Fermi systems, the Fermi level defines hole and particle sp states. Most of the correlations take place close to the Fermo level

10. p-h DMRG basics F Let's consider for simplicity axially-symmetric Nilsson-like levels, which admit four states (s=4):

11. When we add the next level: number of particle states goes from m to s×m number of hole states goes from m to s×m number of states involving particles coupled to holes also goes up. F

12. … F

13. … F

14. Basic idea of DMRG method: F truncate from the s×m states for particles to the optimum m of them, and likewise from the s×m states for holes to the optimum m of them.

15. Finite procedure medium environment superblock m x s x m starting point: infinite procedure size of superblock and medium stay the same while environment block shrinks medium block stored from previous iteration “zipping” back and forth → iterative convergence warm up 1 st sweep2 nd sweep

16. Sampling criterion: FAQ 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 HamiltonianDiagonalize the Hamiltonian

17. … Define the reduced density matrices for particles and holesDefine the reduced density matrices for particles and holes Diagonalize these matrices:Diagonalize these matrices: represent the probability of finding a particular  -state in the full ground state wave function of the system; represent the probability of finding a particular  -state in the full ground state wave function of the system;

18. … Optimal truncation corresponds to retaining a fixed number of eigenvectors that have largest probability of being in ground state, i.e., have largest eigenvalues; Parameter of the procedure: number of states retained after each interaction; 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;

19. Must calculate matrix elements of all relevant operators at each step of the procedureMust calculate matrix elements of all relevant operators at each step of the procedure The highest memory consuming operators within a block areThe highest memory consuming operators within a block are They can be contracted with the interaction and be reduced to O(1) and O(L)They can be contracted with the interaction and be reduced to O(1) and O(L) Subtleties:

20. 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.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 density matrices that simultaneously include info on several states of the system.Procedure as described guarantees optimization of ground state. To get optimal description of many states, we may need to construct density matrices that simultaneously include info on several states of the system. Legeza and Solyom used quantum information concepts like block entropy and entanglement to conclude that the DMRG is extremely sensitive to the level ordering and the initialization procedure.Legeza and Solyom used quantum information concepts like block entropy and entanglement to conclude that the DMRG is extremely sensitive to the level ordering and the initialization procedure.

21. ph-DMRG: model calculations –Hamiltonian –40 particles in j=99/2 shell –size of the superblock n dim ~10 26 –parameters:

22. ph-DMRG: realistic nuclear structure calculations Hamiltonian

23. ph-DMRG: realistic nuclear structure calculations configuration space

24. ph-DMRG: 24 Mg in m - scheme  sd-shell  4 valent protons  4 valent neutrons  USD interaction Sph HF

25. ph-DMRG: Infinite vs. finite procedure

26. ph-DMRG: 48 Cr in the j-scheme

27. ph-DMRG: 48 Cr in the m-scheme

28. The Oak Ridge DMRG program Thomas Papenbrock from ORNL developed an alternative program for doing nuclear structure calculations with the DMRG: DMRG with sweeping in the m-scheme Axial HF basis. The levels from the Fermi energy. In the warm up, protons are the medium for neutrons and vice versa. In the sweeping, protons are to the left of the chain and neutrons to the right.

29. The Oak Ridge DMRG program T. Papenbrock and D. J. Dean, J.Phys. G 31 (2005) S1377 56 Ni

30. Comments about the results & Outlook By working in symmetry broken basis we did not preserve angular momentum. Conservation of angular momentum would require: Work in spherical single particle basis Include all states from a given orbit in a single shot. Avoid truncations within a set of degenerate density matrix eigenvalues.

31. Comments about the results & Outlook Include sweeping (results do not show expected improvement) Apply an effective interaction theory that renormalizes the interaction within the superblock space.