Open problems in the interpretation of the observed pair of near-degenerate ΔI = 1 bands with the same parity as the chiral doublet bands are discussed. The ambiguities for the existing fingerprints of the chirality in atomic nuclei and problems in existing theory are discussed, including the description of quantum tunneling in the mean field approximation as well as the deformation, core polarization and configuration of the particle rotor model (PRM). Future developments of the theoretical approach are anticipated.

https://iopscience.iop.org/article/10.1088/0954-3899/37/6/064025/pdf

Open problems in understanding the nuclear chirality

Open problems in the interpretation of the observed pair of near degenerate $\Delta I = 1$ bands with the same parity as the chiral doublet bands are discussed. The ambiguities for the existing fingerprints of the chirality in atomic nuclei and problems in existing theory are discussed, including the description of quantum tunneling in the mean field approximation as well as the deformation, core polarization and configuration of particle rotor model (PRM). Future developments of the theoretical approach are prospected.

Open Problems in Understanding the Nuclear Chirality

Our understanding of angular-momentum-projection goes beyond quantum-number restoration for symmetry-violated states. The angular-momentum-projection method can be viewed as an efficient way of truncating the shell-model space which is otherwise too large to handle. It defines a transformation from the intrinsic system, where dominant excitation modes in the low-energy region are identified with the concept of spontaneous symmetry breaking, to the laboratory frame with well-organized configuration states according to excitations. An energydictated, physically-guided shell-model truncation can then be carried out within the projected space and the Hamiltonian is thereby diagonalized in a compact basis. The present article reviews the theory of angular-momentum-projection applied in the nuclear many-body problem. Angular momentum projection emerges naturally if a deformed state is treated quantum-mechanically. To demonstrate how different physical problems in heavy, deformed nuclei can be efficiently described with different truncation schemes, we introduce the projected shell model and show examples of calculation in a basis with axial symmetry, a basis with triaxiality, and a basis with both quasiparticle and phonon excitations. Technical details of how to calculate the projected matrix elements and how to build a workable model with the projection techniques are given in the appendix.

https://iopscience.iop.org/article/10.1088/0031-8949/91/4/043005/pdf

Projection techniques to approach the nuclear many-body problem

Recent high-spin observations reveal entirely new modes of collective rotational motion and the existence of novel symmetries and spontaneous symmetry breaking phenomena, uncovering hitherto unexploited coupling schemes of intrinsic and collective degrees of freedom. It continues to stimulate the theoretical progress in the field, which clearly turns towards a microscopic description based on self-consistent approaches using either an effective non-relativistic Hamiltonian or an effective relativistic Lagrangian. New coupling schemes call not only for symmetry unrestricted mean-field theories, but also for extensions going beyond the mean-field. The progress in the development of these theoretical methods is discussed in this review.

https://iopscience.iop.org/article/10.1088/0034-4885/68/1/R03/pdf

Mean-field description of high-spin states

The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Examples are strongly interacting atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. However, mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group theoretical techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. We present applications of the projection methods to simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the energy density functional theories are highlighted.

/pdf/symmetry-restoration-in-mean-field-approaches-5ae1s34kug.pdf

Symmetry restoration in mean-field approaches

The projected shell model analysis is carried out using the triaxial Nilsson $+$ BCS basis. It is demonstrated that, for a better description of the moments of inertia of nuclei in the transitional region, it is necessary to take the triaxiality into account and perform the three-dimensional angular momentum projection from the triaxial Nilsson $+$ BCS intrinsic wave function.

K. Hara

Papers

Triaxial projected shell model approach