The Shell Model and the DMRG Approach. Stuart Pittel Bartol Research Institute and Department of Physics and Astronomy, University of Delaware. Introduction.
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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