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International Journal of Modern Physics E

MAXIMAL-DECOUPLING VARIATIONAL PRINCIPLE AND OPTIMAL AUXILIARY HAMILTONIANS FOR NUCLEAR COLLECTIVE MOTIONS

Ching‐Teh Li

Published Jan 1, 1993• 0 citations
Publisher

Abstract

A variational principle of maximal decoupling between the collective and noncollective subspaces has recently been proposed to pin down the collective degrees of freedom of the nuclear many-body problem. This maximal-decoupling variational principle is further utilized here to determine in an optimal way the auxiliary Hamiltonians to be used in severely truncated boson-space calculations for nuclear collective motions. In the present method one treats the spurious-state problem in truncated boson-space calculations by using appropriate auxiliary boson Hamiltonians. The viability of the method is demonstrated through its application to a simplified shell model containing only the monopole pairing interactions.

DOI
10.1142/s0218301393000534

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A variational principle of maximal decoupling between the collective and noncollective subspaces has recently been proposed to pin down the collective degrees of freedom of the nuclear many-body problem. This maximal-decoupling variational principle is further utilized here to determine in an optimal way the auxiliary Hamiltonians to be used in severely truncated boson-space calculations for nuclear collective motions. In the present method one treats the spurious-state problem in truncated boson-space calculations by using appropriate auxiliary boson Hamiltonians. The viability of the method is demonstrated through its application to a simplified shell model containing only the monopole pairing interactions.
A new method based on a maximal-decoupling variational principle is proposed to treat the Pauli-principle constraints for calculations of nuclear collective motion in a truncated boson space. The viability of the method is demonstrated through an application to the multipole form of boson Hamiltonians for the single-j and nondegenerate multi-j pairing interactions. While these boson Hamiltonians are Hermitian and contain only one- and two-boson terms, they are also the worst case for truncated boson-space calculations because they are not amenable to any boson truncations at all. By using auxiliary Hamiltonians optimally determined by the maximal-decoupling variational principle, however, truncations in the boson space become feasible and even yield reasonably accurate results. The method proposed here may thus be useful for doing realistic calculations of nuclear collective motion as well as for obtaining a viable interacting-boson-model type of boson Hamiltonian from the shell model.
The configuration-interaction shell model is an effective and widely-used approach to the nuclear many-body problem, whose main drawback is the exponential growth of the basis dimension. An useful way to character nuclear shell-model Hamiltonians is through traces, including traces in subspaces defined by orbital occupations. Such traces, or energy centroids, can be easily and efficiently computed through the monopole components of the nuclear interaction, that is, terms that go like nanbn_a n_bna​nb​ where nan_ana​ is the occupation of the single-particle orbital labeled by aaa. These calculations can be carried out very quickly for both empirical (valence space) and no-core shell model spaces and interactions. In fact, they can be carried out so fast, one can use this to optimize an efficient, if approximate, many-body truncation scheme used in available nuclear shell-model codes such as BIGSTICK. To carry out both the traces and the optimization, we present the TRACER code, written in Fortran90 and described and available here. We give example results as well as discuss performance.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson systems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.
The time-dependent variational principle of the Schr\"odinger equation is applied to a formally exact solution of the Schr\"odinger equation whose variational elements are operators which define a collective subspace of the many-body system under study. This generalizes the procedure employed to derive time-dependent Hartree-Fock theory. Four distinct formally exact time-independent variational principles, including several familiar forms, are derived. Application to a class of exactly soluble models, studied in the vibrational regime, illustrates the different modes of implementation of two of the principles. The general theory of large amplitude collective motion is derived. It is shown that the principles can be applied equally well, using either boson or fermion pair degrees of freedom. Some aspects of the relation to the semiclassical limit are discussed.[NUCLEAR STRUCTURE Time-dependent variational principle for collective motion. Invariance principle of Schr\"odinger equation. Generalized coherent state trial function. Exact time-independent variational principles. Variational principle for intrinsic state. Cranking variational principle. Large amplitude collective motion. Semiclassical limit.]