What is the collective subspace in the many-nucleon Hilbert space? Original Research Article

The shell-model calculation in the full shell-model Hilbert space of a many-nucleon system is, in general, very difficult when the nucleon number is large. For such a large-dimensional system, we have to truncate the Hilbert space to a small size of collective subspace. To do this, we carry out the Dyson-type boson mapping and truncate the boson degrees of freedom to some small number of collective bosons; for example, s-, d- and g-bosons only. Thereby we can divide the Dyson boson Hamiltonian into the collective part and the non-collective part; the latter includes the renormalization of the effects from the non-collective degrees of freedom. Comparing numerical calculations under this boson approximation with the exact shell-model calculations, we show that our truncation gives a very good approximation to the exact shell model. Thus we can study the meaning of the collective subspace and establish the foundation of the basic idea of the interacting boson model (IBM).