Structural analysis of a collective subspace in the dynamical generator coordinate method

Abstract

In nuclear theory, the generator coordinate method (GCM), a type of configuration mixing method, is often used for the microscopic description of collective motions. However, the GCM has a problem that a structure of the collective subspace, which is the Hilbert space spanned by the configurations, is not generally understood. In this paper, I investigate the structure of the collective subspace in the dynamical GCM (DGCM), an improved version of the GCM. I then show that it is restricted to a specific form that combines tensor products and direct sums under reasonable conditions. By imposing additional specific conditions that are feasible in actual numerical calculations, it is possible to write the collective subspace as a simple tensor product of the collective part and the others. These discussions are not dependent on the details of the function space used for generating the configurations and can be applied to various methods, including the mean-field theory. Moreover, this analytical technique can also be applied to a variation after projection method (VAP), then which reveals that under a specific condition, the function space of the VAP has an untwisted structure. These consequences can provide powerful tools for discussing the collective motions with the DGCM or the GCM.

Applications of the dynamical generator coordinate method to quadrupole excitations

“…For example, if we take the cut-off of 10 −10 , the sum rule for the excited states remain unsatisfactory in the GCM, whereas that in the DGCM is good up to the seventh excited state. This is related to the fact that under certain ansatze the collective subspace in the DGCM can be represented as a tensor product of the collective and the noncollective degrees of freedom [36].…”

“…Therefore, we shall use this method in this paper as well. Considering the DGCM constructed in this way, one can show that under certain conditions the collective subspace defined with the DGCM has good properties [36]. As a result, an unexpected quantum entanglement is less likely to occur in the states obtained as solutions of the Hill-Wheeler equation.…”