Dear Colleagues,

Mean-field models have played a pivotal role in elucidating the fascinating properties of the nuclear many-body system. The most straightforward mechanism known to incorporate correlations in a quantal many-body system is employing a mean-field or a product wavefunction that spontaneously breaks the symmetries obeyed by the Hamiltonian of the system. These broken symmetries include rotational, isospin, reflection, and gauge symmetry associated with the particle number. The restoration of the symmetries is essential for evaluating observable quantities, which can then be compared with the experimental data. The experimental data are now becoming available at the extremes of isospin and angular momentum, with the availability of radioactive beam facilities and the development of state-of-the-art detection systems. Fluctuations play a crucial role in these extreme situations, and beyond mean-field description, using symmetry restored methods becomes imperative. Although symmetry projection methods have been well known for over fifty years, application of these methods to realistic Hamiltonians and model spaces poses several challenges. The primary objective of the present Special Edition is to deliberate on these impediments with all technical details and to provide a possible roadmap for the resolution of these problems. This Special Edition will also invite proposals on the application of symmetry restoration methods to unbound systems where the coupling between discrete and continuum spaces becomes essential.

Prof. Dr. Javid Sheikh 
Prof. Dr. Peter Ring
Guest Editors

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• mean-field models
• spontaneous symmetry breaking
• symmetry restoration methods
• rotational, isospin, and gauge symmetries
• unbound systems
• projection methods in the continuum

Title: Isospin symmetry breaking in atomic nuclei
Authors: J. A. Sheikh; R. N. Ali; Praveen Chandra Srivastava
Affiliation: 1. Department of Physics, University of Kashmir, Hazratbal, Srinagar, 190 006, India; 2. Department of Physics, Central University of Kashmir, Tulmulla, Ganderbal, 191131, India 3. Department of Physics, Indian Institute of Technology Roorkee, Roorkee, 247667, India
Abstract: The importance of the isospin symmetry breaking in elucidating the properties of atomic nuclei is reviewed. The experimental data on isobaric analogue states cannot be described only with the Coulomb interaction and isospin breaking terms in the nucleon-nucleon interaction are needed to discern the observed properties. In the present work, isospin symmetry breaking terms are explicitly considered in nuclear energy density functional and spherical shell model approaches, and a detailed investigation of the analogue states and other isospin sensitive modes of nuclei is performed. Furthermore, isospin mixing of the nuclear states is studied with the explicit projection of the isospin quantum number.

Title: Symmetry restoration in an arbitrary spatial domain
Authors: L. M. Robledo
Affiliation: Departamento de Fısica Teorica and CIAFF, Facultad de Ciencias, Universidad Aut ́onoma de Madrid, E-28049 Madrid, Spain
Abstract: Symmetry restoration has traditionally dealt with whole systems living in three dimensional space. However, in the theoretical description of fission the initial whole system scissions into two or more fragments that are well defined in a given spatial domain. The wave function of the fragments should also include the appropriate quantum numbers in their description and therefore symmetry restoration has to be extended to this case. Disentangling fragment's wave functions is not possible in general and therefore one has to act on the total wave function but restricting the symmetry restoration procedure to a given spatial domain encompassing the fragment. The implementation of the new constraint requires different techniques than the traditional ones and also leads to some difficulties associated to dealing with a non-complete Hilbert space. How to generalize the symmetry restoration methods to incomplete Hilbert spaces shall be the main theme of the present work?

Title: Pseudospin symmetry and the hints for unstable and superheavy nuclei
Authors: Jing Geng; Zhi Heng Wang; Jia Liu; Jia Jie Li; Wen Hui Long
Affiliation: School of Nuclear Science and Technology, Lanzhou University
Abstract: The pseudo-spin symmetry (PSS) provides an important angle to understand nuclear microscopic structure and the novel phenomena found in unstable nuclei. The relativistic Hartree-Fock (RHF) theory, that takes the important degrees of freedom associated with the $\pi$-meson and $\rho$-tensor ($\rho$-T) couplings into account, provides appropriate description on the PSS restoration in realistic nuclei, particularly for the pseudo-spin (PS) doublets with high angular momenta ($\tilde l$). The investigations of the PSS within the RHF theory are recalled in this paper, by focusing on the effects of the Fock terms. Aiming at common artificial shell closures appearing in previous relativistic mean field calculations, the mechanism responsible for the PSS restoration of high-$\tilde l$ orbits are stressed, that reveals the manifestation of nuclear in-medium effects on the PSS, and thus provides qualitative guidance on modeling the in-medium balance between nuclear attractions and repulsions. Moreover, the essential role played by the $\rho$-T coupling, that contributes mainly via the Fock terms, is introduced as combined with the relations between the PSS and various nuclear phenomena, including the shell structure and the evolution, novel halo and bubble-like phenomena, and the superheavy magicity. As the consequences of nuclear force in complicated nuclear many-body systems, the PSS itself and the mechanism therein can not only deepen our understanding on nuclear microscopic structure and relevant phenomena, but also provide special insight into the nature of nuclear force, which can further enrich our knowledge of nuclear physics.