The Shell Model and the DMRG Approach. Stuart Pittel Bartol Research Institute and Department of Physics and Astronomy, University of Delaware. Introduction.
Bartol Research Institute and Department of Physics and Astronomy, University of Delaware
- Introduced by Steven White in the early 90s to treat quantum lattices.
[S. R. White, PRL 69, 2863 (1992); S. R. White, PRB48, 10345 (1993); S. R. White and D. A. Huse, PRB48, 3844 (1993).]
- J. Dukelsky and SP, The density matrix renormalization group for finite fermi systems, J. Dukelsky and S. Pittel, Rep. Prog. Phys. 67 (2004) 513.
and store them.
where t denotes the number of states in the medium.
Ground state density matrix for the enlarged block is then constructed and diagonalized.
Truncate to the p eigenstates with largest eigenvalues. By definition, they are the most important states of the enlarged block in the ground state of the superblock, i.e. the system.
- After going thru all layers, reverse direction and update the blocks based on results stored in previous sweep. Done iteratively until acceptably small change from one sweep to the next.
- Requires a first pass, called the warmup stage. Here we could, e.g., use the Wilson RG method to get a first approximation to the optimum states in each block. Since they will be improved in subsequent sweeps, not crucial that it be very accurate approximation.
- Partition neutron versus proton orbitals. Neutron orbits on one side of the “chain” and proton orbits symmetrically on the other.
- Use orbits that admit two particles (nlj+m and nlj-m).
- Such an m-scheme approach violates angular-momentum conservation, which may be severe if truncation is significant.
- Order the orbits so that most active (i.e., those nearest the Fermi surface) are at the center of the chain. This is based on work of Legeza and collaborators.
- Use closed shell plus 1p-1h states to define output from warmup phase.
Also did calculations for 56Ni, but results not as good.
(1) Model space;
(2) number of active neutrons and protons;
(3) shell-model H;
(4) single-shell reduced matrix elements for all active orbits and all sub-operators of H.
- EGS=-180. Complete basis of 0+ states has 158 states.
- Warmup gives EGS=-180 with all 158 states.
- Any number of sweeps give the same results since full space always used.
Results for p=10: enlargement.
- After first sweep, obtain EGS=-180 with a basis of 38 states
- After first sweep, getEGS=-180 with a basis of 32 states