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arXiv:2205.13058

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

N. Hizawa

Published May 25, 2022• 0 citations
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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.

DOI
10.48550/arxiv.2205.13058

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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.
A procedure is developed for treating a nuclear collective motion via the path integral formalism by defining a collective propagator in the "phase space" of the collective subspace. This collective subspace is projected out of the full many-body Hilbert space by an approach of the Generator Coordinate Method. Under the approximation of quadratic "momentum" dependence we write the propagator in the coordinate representation from which we derive the collective Schrödinger equation in terms of a canonical pair and .
A general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL + (3,R) and the quotient set GL + (3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. It is proposed that an appropriate basis function for the GL + (3,R) collective model is provided by an irreducible representation of the boson SU(6) group.
A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the 4He^{4}\mathrm{He}4He nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.
The Goldstone-Brueckner perturbation theory is extended to incorporate in a simple way correlations associated with large-amplitude collective motions in nuclei. The new energy expansion making use of nonorthogonal vacua still allows to remove the divergences originating from the hard core of the bare interaction. This is done through the definition of a new Brueckner matrix summing generalized Brueckner ladders. At the lowest order, this formalism motivates variational calculations beyond the mean field, such as the generator coordinate method (GCM) and the projected mean-field method, from a perturbative point of view for the first time. Going to higher orders amounts to incorporate diabatic effects in the GCM and to extend the projection technique from product states to well-defined correlated states.
It is the purpose of the present paper to illustrate the quantum theory of dynamical collective subspace, which has been proposed in a previous paper to describe the large-amplitude collective motion, by using a simple four-level model with the pairing interactions. The quantum theory provides us a prescription to dynamically specify a set of "global" collective operators as well as the "optimum" dynamical collective subspace spanned by them. This prescription is applied to the simple model, and the results thus obtained are compared with exact numerical solutions. It is shown that the results by our theory are in excellent agreement with the exact solutions, not only in the low-lying collective states in the vicinity of the "vacuum" with the stable mean-field but also in the higher excited collective states far from it. Through the investigation, it is demonstrated that our theory is quite powerful in disclosing microscopic structure of transfiguration of collectivity near the critical point of the "phase transition". It is the essential statement derived by our theory that the conventional (local) collective operators based on the Tamm-Dancoff approximation or the random phase approximation should be replaced by relevant (global) collective operators with changeable microscopic structure depending on the amplitude of collective motion, in order to properly specify a set of collective excited states.
This paper discusses quantum algorithms for the generator coordinate method (GCM) that can be used to benchmark molecular systems. The GCM formalism defined by exponential operators with exponents defined through generators of the fermionic U(N)U(N)U(N) Lie algebra (Thouless theorem) offers a possibility of probing large subspaces using low-depth quantum circuits. In the present study, we illustrate the performance of the quantum algorithm for constructing a discretized form of the Hill-Wheeler equation for ground- and excited-state energies. We also generalize the standard GCM formulation to multiproduct extension that when collective paths are properly probed can systematically introduce higher rank effects and provide elementary mechanisms for symmetry purification when generator states break the spatial or spin symmetries. The GCM quantum algorithms also can be viewed as an alternative to existing variational quantum eigensolvers, where multistep classical optimization algorithms are replaced by a single-step procedure for solving the Hill-Wheeler eigenvalue problem.