In the nuclear shell model, the low-energy structure of a nucleus is traditionally described by assuming an inert doubly–magic core and diagonalizing the effective nuclear Hamiltonian in an active space involving at most a few major shells. Despite the enormous truncation inherent in this approach, the method can still only be applied in very limited nuclear regimes. For sufficiently heavy nuclei, for example, further truncation further is required to reduce the number of shell-model configurations to a manageable size.
An attractive truncation possibility is provided by the Density Matrix Renormalization Group (DMRG), a method initially developed for low-dimensional quantum lattices, and later extended to finite Fermi systems. In the latter context, it has been applied to the description of small metallic grains, to problems in quantum chemistry and to two-dimensional electrons in strong magnetic fields. The successes achieved in these various applications suggests that it might also prove useful as a dynamical truncation strategy for obtaining accurate approximate solutions to the nuclear shell model.
The DMRG method involves a systematic inclusion of the degrees of freedom of the many-body problem. When treating quantum lattices, real-space sites are added iteratively. In finite Fermi systems, these sites are replaced by single-particle levels. At each stage, the system [referred to as a block] is enlarged to include an additional site or level. This enlarged block is then coupled to the rest of the system (the medium) giving rise to the superblock. For a given eigenstate of the superblock (often the ground state) or perhaps for a group of important eigenstates, the reduced density matrix of the enlarged block in the presence of the medium is constructed and diagonalized and those states with the largest eigenvalues are retained.
This process of systematically growing the system and determining the optimal structure within that enlarged block is carried out iteratively, by sweeping back and forth through the sites, at each stage using the results from the previous sweep to define the medium. In this way, the process iteratively updates the information on each block until convergence from one sweep to the next is achieved. Finally, the calculations are carried out as a function of the number of states retained in each block, until the changes are acceptably small.
The traditional DMRG method, when applied in nuclei and elsewhere, works in a simple product space, whereby the enlarged block is obtained as a product of states in the block and the added site and likewise the superblock is obtained as a product of states in the enlarged block and the medium. In the context of nuclear terminology, this is equivalent to working in the m-scheme. A limitation of the traditional algorithm is that it does not preserve symmetries throughout the iterative enlargement process. Since the density matrix procedure involves a truncation at each of the iterative stages, there is a potential to lose these symmetries and the associated correlations. On this basis, we proposed the adoption of a strategy whereby angular momentum is preserved throughout the iterative DMRG process.
This method, called the JDMRG, was applied in nuclear physics for the first time in the context of the Gamow Shell Model. It was subsequently developed for application to the traditional shell model by Pittel and Sandulescu, where a first test application to 48Cr was reported. We have now dramatically improved the JDMRG algorithm, to the point where it can be applied to significantly heavier nuclei. In this presentation, we report test results for the largest calculations carried out to date using this method, for the nucleus 56Ni.
B. Thakur, S. Pittel, and N. Sandulescu, Density matrix renormalization group study of 48Cr and 56Ni, Phys. Rev. C 78, 041303(R) (2008). [PDF]