*3.2. Quasiparticle Vacua Shell Model*

The MCSM wave function is a linear combination of the Slater determinants, which are not suitable for treating the pairing correlation in the heavy-mass region. In order to include the pairing correlation efficiently, one can replace the Slater-determinant basis by the number projected quasi-particle vacua. This framework is called the "QVSM" [24]. The QVSM wave function is defined as

$$|\Psi^{(N\_b)}\rangle = \sum\_{n=1}^{N\_b} \sum\_{K=-J}^{J} f\_{iK} P\_{MK}^{J\pi} P^Z P^N |\phi\_n\rangle \tag{10}$$

where *<sup>P</sup><sup>Z</sup>* and *<sup>P</sup><sup>N</sup>* are the proton and neutron number projectors, respectively. |*φn* is a quasiparticle vaccum and is given as

$$\begin{aligned} \left| \mathcal{J}\_k^{(n)} \right| \left| \phi\_n \right\rangle &= \begin{array}{c} 0 \quad \text{for any } k, \\ \mathcal{J}\_k^{(n)} &= \sum\_i (V\_{ik}^{(n)\*} c\_i^\dagger + \mathcal{U}\_{ik}^{(n)\*} c\_i), \end{array} \tag{11}$$

where *β<sup>k</sup>* denotes a quasi-particle annihilation operator and *c*† *<sup>i</sup>* is the creation operator of the single-particle orbit *i*. Thus, the basis state is parametrized by the complex matrices *U* and *V* which keep the orthogonalization condition [78]. In the present work, this quasiparticle does not mix the proton and neutron spaces. The energy is obtained in the same manner to the MCSM by solving Equation (9). *U*(*n*) and *V*(*n*) matrices are determined utilizing the conjugate gradient method to minimize *E*(*n*).

As a benchmark test, shell-model calculations of 132Ba were performed with the SN100PN interaction [79] and the *jj*55 model space. Figure 3 represents the results of various approximation frameworks. In Figure 3, (a)–(d) present the results of the generator coordinate methods (GCM) discussed in [58] in comparison with the MCSM result (e), the QVSM result (f), and the exact shell-model energy (g). Figure 3A shows that the two GCM results with the Slater determinants, (a) and (b), have 2 MeV or a larger deviation from the exact one. The GCM results with the quasiparticle vacua basis, the MCSM result, and the QVSM result show the deviation smaller than 1 MeV. The QVSM is the best result, the deviation of which is only 45 keV.

**Figure 3.** Shell-model results of 132Ba with the SN100PN interaction by various approximation methods. (**A**): the difference of shell-model energies from the exact one. (**B**): the excitation energies of the ground-state band (2<sup>+</sup> <sup>1</sup> , 4<sup>+</sup> <sup>1</sup> , 6<sup>+</sup> <sup>1</sup> and 8<sup>+</sup> <sup>1</sup> ) and the quasi-*<sup>γ</sup>* band (2<sup>+</sup> <sup>2</sup> , 3<sup>+</sup> <sup>1</sup> , 4<sup>+</sup> <sup>2</sup> , 5<sup>+</sup> <sup>1</sup> , and 6<sup>+</sup> <sup>2</sup> ). The results are shown for: (a) generator-coordinate method (GCM) with the Hartree-Fock (HF) calculations assuming axial symmetry, (b) GCM with the HF calculations without assuming axial symmetry, (c) GCM with the Hartree-Fock-Bogoliubov (HFB) calculations assuming axial symmetry, (d) GCM with the HFB calculations without assuming axial symmetry, (e) MCSM with 50 basis states without variance extrapolation, (f) QVSM with 30 basis states without variance extrapolation, and (g) exact shell-model result by the Lanczos diagonalization. Numerical data of (a–d) and (g) are taken from Ref. [58].

Figure 3B shows the excitation energies of the yrast band (2<sup>+</sup> <sup>1</sup> , 4<sup>+</sup> <sup>1</sup> , 6<sup>+</sup> <sup>1</sup> , and 8<sup>+</sup> <sup>1</sup> ) and the quasi-gamma band (2<sup>+</sup> <sup>2</sup> , 3<sup>+</sup> <sup>1</sup> , 4<sup>+</sup> <sup>2</sup> , 5<sup>+</sup> <sup>1</sup> , and 6<sup>+</sup> <sup>2</sup> ) of 132Ba. The two GCM methods with the Slater determinant basis states, (a) and (b), present too low excitation energies of the yrast band because of the underestimation of the pairing correlation. The two GCM methods with the quasiparticle vacua basis, (c) and (d) show the correct excitation energies of the yrast band. Two GCM methods assuming axial symmetry, (a) and (c), apparently failed to reproduce the quasi-gamma band. The GCM method with the quasiparticle vacua basis is referred to as the HFB+GCM ("HFB" stands for "Hartree-Fock-Bogoliubov") in Figure 3. The HFB+GCM result shows reasonable agreement with the exact one. The MCSM also shows the reasonable agreement with the exact one, the moment of inertia is slightly overestimated because of the underestimation of the pairing correlation. The QVSM result shows the almost perfect agreement with the exact one.

This benchmark test confirms that the QVSM outperforms the GCM and the MCSM in this mass region. Indeed, the QVSM wave function is superior to the MCSM wave function of the same *Nb* by including the pairing correlation efficiently. However, the computation time of the QVSM is longer than the MCSM with the same *Nb* mainly due to the number projection. If the difference between the MCSM and QVSM energies with the same *Nb* is small, the MCSM can surpass the QVSM by increasing *Nb* within the same computation time. In Ref. [24], we demonstrated that in the case of 68Ni in the *pfg*9/2*d*5/2 model space the MCSM and QVSM energies have small deviation with the same *Nb*, and thus the MCSM is efficient in terms of the computation time. In practice, one can try both the QVSM and the MCSM with a small *Nb* and identify which one is efficient before performing heavy calculations. Empirically, it was found that the QVSM is more efficient in nuclei heavier than tin isotopes, in which the pairing correlation becomes important.

As another test of the QVSM, one can perform shell-model calculation of 101Sn with the 0*g*9/2, 0*g*7/2, 1*d*5/2, 1*d*3/2, and 2*s*1/2 orbits as the model space. One adopts an effective interaction derived in an ab initio way, the VS-IMSRG method [15,80]. In the derivation, the chiral N3LO 1.8/2.0(EM) (see details in Ref. [81]) was adopted for the two-body and three-body forces with a similarity-renormalization-group evolution. Figure 4 shows the results of the 7/2<sup>+</sup> <sup>1</sup> and 5/2<sup>+</sup> <sup>1</sup> energies of 101Sn provided by ITSM (Figure 4a), the Lanczos diagonalization with particle-hole truncation (Figure 4b), and the QVSM (Figure 4c).

**Figure 4.** Energies of the 5/2<sup>+</sup> (red) and 7/2<sup>+</sup> (black) states of 101Sn. (**a**): ITSM result against *T*max from Ref. [80]. (**b**): Result of the Lanczos diagonalization with *t*-particle *t*-hole truncation. (**c**): QVSM with 60 basis states. The solid lines in (**c**) are fitted for the extrapolation. See text for details.

Figure 4a shows the ITSM result as a function of the number of the allowed particlehole excitation across the *N* = *Z* = 50 gap, *T*max [80]. The result shows good convergence as a function of *T*max and predicts the ground 7/2<sup>+</sup> state and the small excitation energy of the 5/2<sup>+</sup> state. Note that it does not mean a variational upper limit, since these results are extrapolated values as a function of the importance measure. Figure 4b shows the Lanczos diagonalization result with restricting *t*-particle *t*-hole excitation across the *N* = *Z* = 50 gap. The *<sup>M</sup>*-scheme dimension of *<sup>t</sup>* = 6 is 9.6 × 109, which is quite large. The energies gradually lower as a function of *t*, but still it does not reach sufficient convergence. Figure 4c presents the QVSM results against the energy variance. as *Nb* increases the energy and energy variance decreases smoothly and approaches the *y*-axis or zero energy variance. The *y*-intercepts of the fitted curve become the extrapolated values, which predict the 7/2<sup>+</sup> ground state with small 5/2<sup>+</sup> excitation energy.

Thus, these three methods predict the 7/2<sup>+</sup> ground state and small excitation energy of the 5/2<sup>+</sup> state consistently. The extrapolated value of the QVSM seems consistent with the behavior of and the diagonalization result with the *t*-particle *t*-hole truncation.

#### **4. Summary**

I reviewed the current status of the shell-model calculations, and our developments to overcome the limitation of the conventional Lanczos diagonalization method. One of the frontiers of shell-model study is to study neutron-rich nuclei towards the neutron drip line, in which a larger model space is required. To perform shell-model calculations with such a large model space, the MCSM was proposed and demonstrated in Section 3.1. The MCSM has been applied to various studies of exotic nuclei [23]. Another frontier is to go heavier open-shell nuclei, in which pairing correlation is essential to be treated efficiently. For such purpose, the QVSM has been developed and its feasibility was demonstrated in Section 3.2. Several benchmark tests to demonstrate the capabilities of the MCSM and QVSM methods are presented.

Other frontiers for the shell-model study are the microscopic description of giant resonance and statistical properties of the highly excited region. The Lanczos strength function method [1] is a solution to this problem, but it is still trapped by the rapid growth of the shell-model dimension. Although several attempts were performed to go beyond this limit, which shows promising results [82], further study is required.

**Funding:** This research was supported by "Program for Promoting Researches on the Supercomputer Fugaku" (JPMXP1020200105) and JICFuS, and KAKENHI grant (17K05433, 20K03981).

**Data Availability Statement:** The data used in this work are available from the author upon reasonable request.

**Acknowledgments:** The author acknowledges Takashi Abe, Michio Honma, Takaharu Otsuka, Takayuki Miyagi, Takahiro Mizusaki, Tomoaki Togashi, Yusuke Tsunoda, and Yutaka Utsuno for fruitful collaborations. I especially thank Takayuki Miyagi for providing us with the VS-IMSRG interaction for 101Sn. This research used computational resources of the supercomputer Fugaku (hp220174, hp210165, hp200130) at the RIKEN Center for Computational Science, Oakforest-PACS supercomputer, Wisteria supercomputer (CCS-Tsukuba MCRP xg18i035 and wo22i022) and Oakbridge-CX supercomputer.

**Conflicts of Interest:** The author declares no conflict of interest.
