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Revista Mexicana de Física

The density matrix renormalization group: a new approach to large-scale nuclear structure calculations

Stuart Pittel, J. Dukelsky

Published Jan 1, 2001• 0 citations
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Abstract

A new approach to large-scale nuclear structure calculations, based on the density matrix renormalization group (DMRG), is described. The DMRG method, which was developed originally in the framework of low-dimensional quantum lattices, has recently been extended to finite Fermi systems. Following a general description of this extended DMRG methodology, we briefly review its first application to a pairing hamiltonian. We then discuss its application to a schematic nuclear model involving many identical nucleons constrained to move in a single large-j shell. In those cases in which exact diagonalization of the hamiltonian is possible, the method is able to reproduce the exact results with extreme precision. Results are also presented for a model problem in which exact solution is not feasible.

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New approach to large-scale nuclear structure calculations

J. Dukelsky, Stuart Pittel

A new approach to large-scale nuclear structure calculations, based on the density matrix renormalization group (DMRG), is described. The method is tested in the context of a problem involving many identical nucleons constrained to move in a single large-$j$ shell and interacting via a pairing plus quadrupole interaction. In cases in which exact diagonalization of the Hamiltonian is possible, the method is able to reproduce the exact results for the ground-state energy and the energies of low-lying excited states with extreme precision. Results are also presented for a model problem in which exact solution is not feasible.

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arXiv:nucl-th/0207025•Jul 11, 2002•1 citation
A new approach to large-scale nuclear structure calculations, based on the density matrix renormalization group (DMRG), is described. The DMRG method, which was developed originally in the framework of low-dimensional quantum lattices, has recently been extended to finite Fermi systems. Following a general description of this extended DMRG methodology, we briefly review its first application to a pairing hamiltonian. We then discuss its application to a schematic nuclear model involving many identical nucleons constrained to move in a single large-j shell. In those cases in which exact diagonalization of the hamiltonian is possible, the method is able to reproduce the exact results with extreme precision. Results are also presented for a model problem in which exact solution is not feasible.
A new approach to large-scale nuclear structure calculations, based on the density matrix renormalization group (DMRG), is described. The method is tested in the context of a problem involving many identical nucleons constrained to move in a single large-jjj shell and interacting via a pairing plus quadrupole interaction. In cases in which exact diagonalization of the Hamiltonian is possible, the method is able to reproduce the exact results for the ground-state energy and the energies of low-lying excited states with extreme precision. Results are also presented for a model problem in which exact solution is not feasible.
The particle-hole density matrix renormalization group $(p\ensuremath{-}h$ DMRG) method is discussed as a possible new approach to large-scale nuclear shell-model calculations. Following a general description of the method, we apply it to a class of problems involving many identical nucleons constrained to move in a single large j-shell and to interact via a pairing plus quadrupole interaction. A single-particle term that splits the shell into degenerate doublets is included so as to accommodate the physics of a Fermi surface in the problem. We apply the $p\ensuremath{-}h$ DMRG method to this test problem for two j values, one for which the shell model can be solved exactly and one for which the size of the Hamiltonian is much too large for exact treatment. In the former case, the method is able to reproduce the exact results for the ground state energy, the energies of low-lying excited states, and other observables with extreme precision. In the latter case, the results exhibit rapid exponential convergence, suggesting the great promise of this new methodology even for more realistic nuclear systems. We also compare the results of the test calculations with those from Hartree-Fock-Bogoliubov approximation and address several other questions about the $p\ensuremath{-}h$ DMRG method of relevance to its usefulness when treating more realistic nuclear systems.
"We briefly review the Density Matrix Renormalization Group (DMRG) method and its potential use in large-scale nuclear shell-model calculations. We propose the use of an angular-momentum-conserving variant of the method (the JDMRG) and report the first test results of such an approach for the nucleus 48 Cr. The positive results of these calculations have motivated us to search for an even more efficient means of implementing the JDMRG strategy and the status of these efforts is also described."
We summarize our recent efforts to develop the Density Matrix Renormalization Group (DMRG) method into a practical truncation strategy for large-scale nuclear shell model calculations. Following an overview of the essential features of the DMRG, we discuss the changes we have implemented for its use in nuclei. In particular, we have found it useful to develop an angular-momentum conserving variant of the method (the JDMRG). We then summarize the principal results we have obtained to date, first reporting test results for 48^{48}48Cr and then more recent test results for 56^{56}56Ni. In both cases we consider nucleons limited to the 2p-1f shell. While both calculations produce a high level of agreement with the exact shell model results, the fraction of the complete space required to achieve this high level of agreement is found to go down rapidly as the size of the full space grows.