# Truncation of large shell-model eigenproblems by model space partitioning

###### Abstract

A method for solving the shell-model eigenproblem in a severely truncated space, spanned by properly selected correlated states obtained by partitioning the full configuration space, is proposed. The method describes in a practically exact way the low energy spectroscopic properties of nuclei, as exemplified in schematic models. The applicability of the method to heavy nuclei as well as in contexts different from the nuclear shell model is stressed.

###### pacs:

21.60.-n 21.60.Cs 74.20.FgComplete shell-model calculations in the region of medium- and heavy-mass nuclei become rapidly prohibitive because of the extremely large configuration space dimensions. Although the increasing power of the present day computers, togheter with a clever usage of the Lanczos algorithm[1], makes feasible exact calculations in configuration spaces of impressive dimensions, the largest complete calculation in a full major shell have been done, as to our knowledge, in the mass region around Ni. The extreme redundancy of the computed quantities is another problem: in the mare magnum of the output of these calculations, one is only interested in the comparison of a limited number of eigenvalues (usually the lowest ones) with the observed energy levels and in the identification of the most significant components of the corresponding wave functions. On the other hand, the truncation of the configuration space may not be an easy task, since taking into account the effect of the excluded space requires a corresponding renormalization of the interaction.

A main road to physically significant reductions of the configuration space is the use of some kind of correlated basis; typical examples are the multi-step shell model[2], the broken-pair approximation[3] or the chain-calculation method[4]. The problem common to these methods which make use of correlated bases is the overcompleteness of the set of basis vectors used in the calculation. Since this redundancy gives rise to spurious admixtures, it is necessary to resort to special techniques to get rid of the spurious components, e.g. computing and analyzing the overlap matrix of the basis, a task which is lenghty and involved.

In this letter we propose a method which accounts in a practically exact way for all the configurations needed to the description of the lower energy states, and is free of all the illnesses of the mentioned truncation schemes. The idea consists in describing the system of interacting nucleons in terms of correlated subsystems defined in orthogonal subspaces. Let us consider a system of valence particles in a given model space, defined by a projection operator

where the refer to all possible independent ways the particles can be distributed over the single-particle (s.p.) levels of the model space. We now partition the s.p. states included in the model space in two groups whose corresponding configuration spaces we call and , so that

where the quantum numbers classify all possible ways of distributing and particles, with , on the s.p. levels of the two partitions, respectively. We accordingly separate the Hamiltonian of the system as

where acts only on the spaces, on the spaces and is the interaction term between and . Solving the eigenvalue problems for all the allowed values of and within each partition separately,

allows to write

(1) |

Eq.(1) is our main point. Since the wave function of the -particle system is written as a direct product of eigenfunctions of correlated subsystems defined in orthogonal spaces, none of the previously mentioned redundancy problems arises, i.e. it combines the advantages of using an orthonormal basis with those offered by a description in terms of correlated subsystems. Shell-model basis is not well suited to further reductions of its dimensions essentially because the residual interaction can be so strong that even configurations which are quite distant in energy from each other may be equally important in the description of a physical state. This is not the case when using correlated bases, where a significant part of the effect of the interaction is already included and even drastic truncations of the basis can be meaningful. As a consequence, an energy criterion to truncate expansion (1) works well: only those basis states which are not too different in energy from the physical state one wants to describe are relevant. It is worth noting that, as is explicitely shown in the following example, there are no particular difficulties in computing the matrix elements of in the basis (1), since they can be expressed in terms of quantities computed in each partition separately.

As a concrete example, let us consider a general two-body shell-model Hamiltonian for a system of identical particles:

(2) |

where and

The partitions yield the two Hamiltonians

(3) |

(4) |

and the interaction term

where and . The matrix elements are defined as

(6) |

and are the particle-hole operators:

(7) |

Let be the eigenstates of for a system of particles distributed over the s.p. levels of the partition 1 of the model space, with corresponding eigenvalues , and the eigenstates of for a system of particles distributed over the s.p. levels of the partition 2, with corresponding eigenvalues .

A complete basis for the configuration space of identical particles, distributed over all the s.p. levels of the model space, with total angular momentum , can be therefore written as

(8) |

where and run over all the possible values such that .

To exemplify the quality of the results obtainable when an energy truncation criterion is used, we consider for simplicity a pairing Hamiltonian

(9) |

where we use the shorthand and and restrict ourselves to a seniority zero approximation for the configuration space. The interaction term (Truncation of large shell-model eigenproblems by model space partitioning) becomes

(10) |

The basis for an -particle system is now The matrix elements of between those basis states are:

where are two-particle transfer amplitudes.

As already pointed out, matrix elements of are quite simple in structure and can be evaluated using quantities defined in each subspace separately. It is worth noting that this is not peculiar of a pairing interaction: as can be easily seen from the structure of Eq.(Truncation of large shell-model eigenproblems by model space partitioning), matrix elements of in the basis (8) can be written in terms of the the two-particles transfer amplitudes , of the particle-hole matrix elements and of the one-particle transfer amplitudes , computed in each partition separately.

To reduce the dimension of the eigenvalue problem we retain only the basis vectors corresponding to values of up to a truncation value : progressively increasing , one can see how the energies in the full space approximate the exact ones. We have chosen a model space of ten s.p. levels divided in two partitions of five s.p. levels. In Table I are reported the first three energies obtained with various truncations of the basis with nucleons. The ten s.p. levels are , which define the partition 1, and , which define the partition 2. The corresponding s.p. energies are, in MeV, 0.0, 3.0, 3.5, 4.0, 5.0, 13.0, 13.5, 15.0, 15.5, 16.0 and we use a constant pairing strength MeV, where . We obtain practically exact results already diagonalizing matrices of order 150 while the dimension of the full space basis is .

Just to show that the existence of a gap in the s.p. energies is not essential to the quality of the approximation, we have also considered the simple, although not at all trivial, example of a pairing problem in a model space of 32 equispaced doubly-degenerate levels. Exact solutions of this problem in the case of 32 particles have been found in Ref.[5] for the energies of the first three states. We arbitrarily choose as partition 1 the first 16 levels and as partition 2 the remaining ones. We have generated almost exact solutions for each value of and within each subspace, with the corresponding two-particle transfer amplitudes, using the method of Refs. [4] [6]. The results, reported in Table II, show an impressive convergence towards the exact values: the best computed energies differ by less than 0.1% from the exact ones, while the corresponding excitation energies, 3.060 and 4.900, are practically exact. It is worth noting that we use a number of states which is an exceedingly small fraction of the full basis dimension . These results are significantly better than those of Ref.[7], where a careful analysis of truncations of the configuration space is made; this fact not only confirms the effectiveness of the present method but also suggests that it can be a viable technique to obtain almost exact solutions for other physically interesting systems like those encountered in the modelization of ultra-small superconducting grains[8][9].

We have considered in the preceding discussion only two partitions, just to be definite. One can have, however, situations where a basis generated by multiple partitioning the configuration space would be required. Althoug we do not treat these cases explicitely in this paper, it is worth noting that the generalization of the method presented here to systems where multiple partitioning is needed is quite straightforward. The method can be naturally applied to systems of neutrons and protons, which can be conveniently treated working out the eigenproblems for neutrons and protons separately by further partitioning of their model spaces, and then diagonalizing the n-p interaction in the product basis; it also constitutes a natural framework to treat excitations outside a major shell, e.g. in the study of ground state correlations.

Acknowledgements We thank N. Lo Iudice for fruitful discussions and comments. This work was supported in part by the Italian Ministero dell’ Università e della Ricerca Scientifica e Tecnologica (MURST).

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