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arXiv:0808.1303 · International Journal of Modern Physics E

THE DENSITY MATRIX RENORMALIZATION GROUP AND THE NUCLEAR SHELL MODEL

Stuart Pittel, B Thakur, Nicolae Sandulescu

Published Dec 1, 2008• 2 citations
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Abstract

We summarize recent efforts to develop an angular-momentum-conserving variant of the Density Matrix Renormalization Group method into a practical truncation strategy for large-scale shell model calculations of atomic nuclei. Following a brief description of the key elements of the method, we report the results of test calculations for 48 Cr and 56 Ni . In both cases we consider nucleons limited to the 2p-1f shell and interacting via the KB3 interaction. Both calculations produce a high level of agreement with the exact shell-model results. Furthermore, and most importantly, the fraction of the complete space required to achieve this high level of agreement goes down rapidly as the size of the full space grows.

DOI
10.1142/s021830130801180x

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We summarize recent efforts to develop an angular-momentum-conserving variant of the Density Matrix Renormalization Group method into a practical truncation strategy for large-scale shell model calculations of atomic nuclei. Following a brief description of the key elements of the method, we report the results of test calculations for 48 Cr and 56 Ni . In both cases we consider nucleons limited to the 2p-1f shell and interacting via the KB3 interaction. Both calculations produce a high level of agreement with the exact shell-model results. Furthermore, and most importantly, the fraction of the complete space required to achieve this high level of agreement goes down rapidly as the size of the full space grows.
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.
We discuss the development of an angular-momentum-conserving variant of the Density Matrix Renormalization Group (DMRG) method for use in large-scale shell-model calculations of atomic nuclei and report a first application of the method to the ground state of 56Ni^{56}\mathrm{Ni}56Ni and improved results for 48Cr^{48}\mathrm{Cr}48Cr. In both cases, we see a high level of agreement with the exact results. A comparison of the two shows a dramatic reduction in the fraction of the space required to achieve accuracy as the size of the problem grows.
We describe the use of the density matrix renormalization group method as a means of approximately solving large-scale nuclear shell-model problems. We focus on an angular-momentum-conserving variant of the method and report test results for the nucleus 48Cr^{48}\mathrm{Cr}48Cr. The calculation is able to reproduce both the ground-state energy and the energy of the first excited state, by diagonalizing matrices much smaller than those of the full shell model.
In this paper, we present systematic results of the J-DMRG method (an angular-momentum-conserving version of the Density Matrix Renormalization Group) in truncated shell-model calculations of nuclei in the 1f – 2p shell. In contrast to earlier studies, these calculations consider both even-even and odd-mass nuclei. Comparison of the results of these calculations with those of exact shell-model treatments of the same nuclei shows that the method converges smoothly to the exact results in all cases and furthermore that the fraction of the full shell-model space required to obtain accurate agreement with the exact results decreases rapidly as the size of the full space grows. This latter result bodes well for the future applicability of the method to even larger nuclei with even larger shell-model spaces as is the ultimate goal of our development of this approach to large-scale shell model studies.
"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."