## Date of Award

Spring 2024

## Degree Type

Open Access Dissertation

## Degree Name

Computational Science Joint PhD with San Diego State University, PhD

## Program

Institute of Mathematical Sciences

## Advisor/Supervisor/Committee Chair

Calvin W. Johnson

## Dissertation or Thesis Committee Member

Fridolin Weber

## Dissertation or Thesis Committee Member

Peter Blomgren

## Dissertation or Thesis Committee Member

Marina Chugunova & Ali Nadim

## Terms of Use & License Information

## Rights Information

© 2024 Ryan M Zbikowski

## Keywords

Computational Physics, Computational Science, Lanczos Algorithm, Nuclear Structure

## Abstract

The nuclear many-body problem is conceptualized in an infinite-dimensional Hilbert space, but computationally solved in a finite one. Thus, the predictive power of microscopic calculations relies on the truncated representation of the infinite-dimensional space as well as leveraging advanced computational methods. My dissertation research focuses on three problems related to computational nuclear physics: exploring aspects of nuclear structure, efficiently solving the large sparse matrix-eigenvalue problem and improving the construction of the many-body basis in the no-core configuration-interaction (NCCI) framework.

I) Elliott’s rotational SU(3) model, and its later extension, the symplectic Sp(3, *R*) model, both played a foundational role in improving the description of nuclear rotational spectral bands. My study of several beryllium isotopes, and 20Ne uses the decomposition of no-core shell-model wavefunctions into symmetry defined subspaces to show the Sp(3, *R*) picture provides a more consistent description of rotational band structure.

II) Solving the non-relativistic many-body Schrödinger equation is often cast as a large sparse Hamiltonian eigenvalue problem. State-of-the-art NCCI calculation dimensions can exceed several billion and typically require supercomputers and thousands of core hours to compute small numbers of low-lying eigenstates. Thus, there is strong motivation for ways to reduce computational costs. In this research, I augment the block Lanczos algorithm using a *bootstrapped* pivot to significantly reduce the number of Hamiltonian-matrix multiplications typically dominating the algorithm’s total time-to-solution. My results demonstrate significant speedup in time-to-solution, often by a factor of two or more, and up to ten, can be achieved through the use of bootstrapping.

III) In NCCI, the many-body basis is historically constructed from antisymmeterized products of harmonic oscillator (HO) single-particle wavefunctions. However, one often needs many HO antisymmeterized products states to produce accurate theoretical predictions of the properties of low-lying nuclear states. Alternative choices of single-particle basis which provide better descriptions of nuclear observables relative to the problem dimension and underlying basis parameters motivate continued explorations. In this research, I explore the use of a natural orbital (NO) single-particle basis, that is, one which diagonalizes the one-body density matrix of a reference many-body state, as a means of improving the description of energy, electromagnetic transitions and radii calculations relative to the problem dimension for select *sd*-shell nuclei.

## ISBN

9798383219454

## Recommended Citation

Zbikowski, Ryan M.. (2024). *Computational Approaches to the Nuclear Many-Body Problem*. CGU Theses & Dissertations, 816. https://scholarship.claremont.edu/cgu_etd/816.