Symmetry projection in atomic nuclei
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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-000111-3
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Review
Symmetry projection in atomic nuclei J.A. Sheikh1,2 and R.N. Ali2,3,a 1 2 3
Cluster University Srinagar, Jammu and Kashmir, Srinagar 190008, India Department of Physics, University of Kashmir, Srinagar 190006, India Department of Physics, Central University of Kashmir, Ganderbal 191131, India Received 12 June 2020 / Accepted 28 August 2020 Published online 23 October 2020 Abstract. In the present work, projection methods employed to restore the spontaneously broken symmetry in the mean-field approach to many-body problem are reviewed. The symmetry restoration is important to include correlations going beyond the mean-field approximation. Further, in order to cause a direct comparison with the experimental quantities, quantum states must have sharp values of the desired quantum numbers. For Hamiltonian based approaches, projection methods have been developed and applied for restoration of symmetries. Symmetry projected Hartree–Fock–Bogoliubov (HFB) equations have been derived in the Hamiltonian based approach, which have a similar structure to those of bare HFB equations. The explicit expressions for the projected Hartree-Fock (HF) and pairing fields are derived in the present work for a generalized projection operator. Approximations to the projection method, based on the work of Lipkin and Kamlah, are discussed with some technical details. Finally, problems associated in the application of the projection technique to the energy density functional approaches are highlighted.
1 Introduction Atomic nucleus is a complex quantum many-body system with number of particles ranging from a few nucleons to a few hundred. Nuclei depict fascinating properties due to strong quantum mechanical shell effects [1]. It has been demonstrated that these shell effects can dramatically change with increasing excitation energy, angular momentum and isospin [2]. In particular, it is now established that shell closures are different far from the valley of stability in lighter and medium mass regions [3]. For an accurate description of shell structures and other nuclear properties, it is imperative to employ realistic nuclear models and methods. The holy grail in nuclear physics are the Ab initio methods aimed at solving the nuclear many-body problem exactly by taking into consideration the realistic nucleon–nucleon (2N, 3N, 4N . . .) interaction [4]. In recent years, the advancements in the computing facilities has made it feasible to apply the ab initio methods that include shell model with no core [5] and Monte-Carlo Green’s function approaches to nuclei up to A = 14 [6]. These ab initio methods are complemented by lattice effective field theory (LEFT) [7], in which the a
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The European Physical Journal Special Topics
nucleons are approximated as point like particles sitting on lattice sites and interacting with each other via the exchange of pions. These complementary m
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