Emergent Phenomena in Atomic Nuclei from Large-Scale Modeling

The following sections are included:

• Preface
• Contents

The collective nature of the nucleus was formalized 63 years ago in the work of Aage Bohr. In other arenas of physics it took 63 years to progress from the “quantum” to the “quark”. Progress in elucidating the details of nuclear collectivity appears to have almost stood still by comparison. The Author gives a personal perspective on nuclear collectivity, using phenomenology and examples from spectroscopy, based on his 50 years as an active nuclear physicist.

With modern computers we can compute nuclear many-body wave functions with an astounding number of components, ≲10¹⁰. But, aside from reproducing and/or predicting experiments, what do we learn from vectors with tens of billions of components? One way to characterize wave functions is through irreducible representations of groups. I discuss briefly the history of group-theoretical characterization of nuclear wave functions, with an emphasis of using Lanczos-type methods to efficiently dissect arbitrary wave functions into group irreps. Although the resulting decompositions are often fragmented over many irreps, one nonetheless finds powerful patterns. First, group decompositions along rotational bands show coherent commonalities, supporting the picture of a shared “intrinsic shape”; this is also called quasi-dynamical symmetry. Second, group decompositions for wave functions using both phenomenological and ab initio forces are often very similar, despite vastly different origins and dimensionalities. Both of these results suggest a group theoretical decomposition can provide a robust “anatomy” of many nuclear wave functions. This in turn supports the idea of using symmetry-based many-body frameworks for calculations.

This Chapter explores the properties of nuclei that are indicators of the emergence of simple dynamics associated with the symmetries that are available to a nucleus as a many-nucleon system. In particular, it focuses on the dynamical symmetries associated with nuclear collective motions. The term “symmetry” is interpreted in a broad sense to mean any set of transformations of the states of a nucleus that leave some important attributes of the nucleus invariant. Symmetries that provide good quantum numbers, or even approximately good quantum numbers, are clearly useful. However, symmetry relations can be used more widely such as, for example, to decompose the states of a nucleus into a sequence of invariant subsets ordered such that their contributions to the observed states of nuclei are of decreasing importance. A standard example of this is given by the spherical harmonic-oscillator shell model which proves to be particularly relevant for many states of near closed shell and singly closed shell nuclei. An alternative, based on the dynamical harmonic-oscillator symmetry which proves to be much more appropriate for the rotational states of deformed nuclei, is reviewed in this paper. A particularly significant result is the emergence of low-energy states of open shell nuclei with the dynamical symmetries of a rotor whose properties depend very little on the details of the nuclear interactions other than that they are basically attractive, of relatively short range, and favor spatially symmetric states.

Basic tenets and selected applications of the symplectic shell model are reviewed. The emphasis is on providing a brief introduction to the structure of the model space and to the tools and techniques needed to carry out model calculations. The case is made that it is necessary and possible to move beyond previously used approximations which employ schematic Hamiltonians and highly truncated model spaces. The rich structure underlying the full symplectic Sp(3, ℝ) model has the potential to shed light on the nature and origin of collective motion in nuclei, and to provide physically relevant truncation schemes that enable large-scale shell-model calculations.

We give a short review of lattice quantum chromodynamics (LQCD) and its status, concentrating on its symmetries and lack thereof. We describe the formalism and motivate the use of large-scale simulations in this field. We pay particular attention to the symmetries intrinsic to the finite-volume discretized lattice, and show how clever applications of the finite-volume symmetries improve numerical simulations. We discuss LQCD calculations related to fundamental symmetries and comment on future prospects of this field from a symmetries point of view.

I review ab initio lattice simulations for nuclear systems using chiral effective field theory. In this discussion we explore the importance of an approximate symmetry of the low-energy nuclear interactions where the four nucleon degrees of freedom can be regarded as four components with an underlying SU(4) symmetry group.

We focus on clustering, especially α-clustering in light nuclei. The nuclear wave function of the cluster model consists of clusterinternal and cluster-relative parts. As the relative motion between the clusters is spatially extended, a shell-model representation of clustering becomes very hard even with present-day large-scale calculations. In contrast to the shell-model basis truncation, the stochastic variational method using a correlated Gaussian basis makes it possible to describe, with no cluster ansatz a priori, both localized cluster states and shell-model-like states as exemplified in ⁴He and ¹⁶O. We also take advantage of the cluster model to calculate electroweak response functions of ^4,6He that involve continuum states.

We review exact and partial symmetries of strongly interacting particles in atomic nuclei. These symmetries serve a key role in facilitating studies of the structure and reactions of isotopes across the nuclear chart, using QCD-inspired interactions and high performance computing (HPC) resources. Namely, they provide a relevant organization of the nuclear model space, while providing an efficient way to reduce the ultra-large model space size through a very structured selection of the basis states to physically relevant subspaces. This enables a deeper understanding of and advancing ab initio studies for determining the microscopic structure of nuclear systems. This, in turn, can guide explorations of simple patterns in nuclei and how they emerge from first principles, as well as extensions of the theory beyond current limitations toward heavier nuclei and larger model spaces.

Auxiliary-field quantum Monte Carlo methods enable the calculation of thermal and ground state properties of correlated quantum many-body systems in model spaces that are many orders of magnitude larger than those that can be treated by conventional diagonalization methods. We review recent developments and applications of these methods in nuclei using the framework of the configuration-interaction shell model.

Density functional theory (DFT) generalizes to Lie DFT, which applies to Lie dynamical groups in many-body quantum mechanics. In Lie DFT, densities are elements of the dual space to a Lie algebra of one-body operators. In this setting, the paper proves an extended Hohenberg–Kohn theorem which guarantees the existence of a dual space energy functional that is minimized by the density of the exact ground state. Lie DFT is relevant to nuclear physics when collective degrees of freedom, say in a deformed isotope, are more significant than an independent fermion picture, like Hartree–Fock or Kohn–Sham mean-field theories. Lie DFT offers a distinct computational advantage: Lie DFT calculations are made in a small finite-dimensional dual space even though the representation space of the dynamical group is large or infinite-dimensional.

Exact solutions of spherical or deformed mean field augmented by various types of pairing are briefly reviewed. Of particular interest are the standard pairing model, the nearest-level pairing model and the extended pairing model. The results for many pairing-driven quantities, such as odd–even mass differences and the moment of inertia, show that these models successfully describe the ground state of spherical nuclei or well-deformed heavy nuclei. Our recent results for level statistics of pair-excitation spectra in selected Ca isotopes described in the standard pairing model, together with the ground-state occupation probabilities of valence nucleon pairs with various angular momenta for even–even ^156-162Yb and groundstate phase transitions in Sm isotopes described by the extended model, are also presented.

The following sections are included:

• Author Index
• Subject Index