**1. Introduction**

Nuclear shell-model calculation is often called the "configuration interaction" (CI) method in analogy to the CI method in quantum chemistry. It provides us with a detailed description of the ground and low-lying excited states of medium-mass nuclei [1]. A shellmodel study is an excellent theoretical tool to discuss exotic structures of nuclei, such as the shape coexistence, shape phase transition, emergence of new magic numbers in neutronrich nuclei, and so on [1,2]. In addition, it enables us to predict nuclear data required for astrophysical applications [3,4] and for elementary particle physics [5–8]. In the shell-model framework, one separates a nuclear wave function into two parts: an inert core and active particles in the model space. Usually, an inert core is taken as a doubly magic nucleus closest to the Fermi level. For example, in the case of 48Ca, 40Ca (*N* = *Z* = 20, where *N* denotes the number of neutrons and *Z* the number of protons in a nucleus) is taken as a frozen inert core and eight valence neutrons are actively occupying the *p f*-shell orbits. Its wave function is written as a linear combination of the Slater determinants each of which represents how the active particles occupy single-particle orbits. This model has achieved successful description of *p*-shell [9], *sd*-shell [10], *p f*-shell, [1,11,12], and *f*<sup>5</sup> *pg*9-shell [13] nuclei with the conventional diagonalization method.

In most of the conventional shell-model studies, the shell-model Hamiltonian is constructed by the many-body perturbation theory [14] with minor phenomenological corrections to fit the experimental data. Its ab initio derivation has been recently developed by the valence-space in-medium similarity renormalization group (VS-IMSRG) method [15], the coupled-cluster method [16], the many-body perturbation theory [17], the extended Krenciglowa–Kuo method [18], and Okubo–Lee–Suzuki approach [19], while ab initio description of strongly quadrupole deformed states is still a challenge [15,20,21]. The shell-model calculation is now applied to ab initio theory and its importance increases now.

However, solving the eigenvalue problem of the shell-model Hamiltonian matrix often requires huge computational resources, which hampers us from performing the shell-model study in the whole nuclear region. Moreover, in the case of neutron-rich nuclei, since the proton and neutron Fermi levels locate in different shells, beyond one-major-shell model space is required for such model space and the exact diagonalization is often infeasible. To

**Citation:** Shimizu, N. Recent Progress of Shell-Model Calculations, Monte Carlo Shell Model, and Quasi-Particle Vacua Shell Model. *Physics* **2022**, *4*, 1081–1093. https:// doi.org/10.3390/physics4030071

Received: 31 July 2022 Accepted: 25 August 2022 Published: 9 September 2022

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circumvent this problem, Tokyo shell-model group introduced the Monte Carlo shell model (MCSM), which provides us with successful description of medium-mass nuclei [22,23]. For heavy nuclei in which the treatment of the pairing correlation is important, the MCSM is not sufficiently efficient and its extension, quasi-particle vacua shell model (QVSM), was introduced [24].

In this paper, I briefly describe the framework of the shell-model calculations and the developments of the shell-model codes in Section 2. In Section 3, various approximation methods going beyond the limitation of the conventional shell-model diagonalization are reviewed. Among them, the MCSM is discussed in Section 3.1. Section 3.2 is devoted to the description of the QVSM framework and its capability. This paper is summarized in Section 4.

#### **2. Conventional Diagonalization Method for Shell-Model Calculations**

In this Section, I briefly review computational aspects of conventional shell-model calculations with the Lanczos method and the developments of shell-model codes.
