Variational Monte Carlo method for shell-model calculations in odd-mass nuclei and restoration of symmetry

Background: While a large-scale shell model calculation (LSSM) is a powerful model to describe the nuclear spectroscopic information, it requires a huge amount of computational resources. As an efficient approximation framework to the LSSM, we have introduced the variational Monte Carlo (VMC) method [T. Mizusaki et al., Phys. Rev. C 85, 021301(R) (2012)]. However, this framework was applicable only to even-mass nuclei.

Purpose: We aim to extend the VMC method for better precision and to make it applicable to odd-mass nuclei.

Methods: We investigate two kinds of extensions for the VMC method with the Pfaffian in the nuclear shell-model calculations. One is the extension to odd-mass nuclei, for which we find a new Pfaffian expression of the VMC matrix elements. The other is the extension of the variation after angular-momentum projection.

Results: We successfully implement the full angular-momentum projected trial state into the VMC method, which can provide us with precise yrast energies. We also find a unique characteristic, namely that this angular-momentum projection in the VMC can be even “approximately” performed.

Conclusions: A unified VMC framework with the variation after projection is given both for even and odd-mass nuclei. The approximate angular-momentum projection is useful not only for efficient computation but also for precise estimation of the yrast energies through the energy-variance extrapolation.

Figure 1

Convergence of energies of $I^{\pi }=0^{+}$ (filled orange circles), $2^{+}$ (open green circles), $4^{+}$ (filled blue squares), $6^{+}$ (open light-blue squares), $8^{+}$ (filled green triangles), ${10}^{+}$ (open brown triangles), and ${12}^{+}$ (filled red inverted triangles) states of ${}^{48}\mathrm{Cr}$ obtained by J-VAP VMC as functions of the number ofe SR iterations. The right column shows the VAP results, exact shell-model energies, and the VBP results of the $I^{\pi }=0^{+}$, $2^{+}$, $4^{+}$, $6^{+}$, $8^{+}$, ${10}^{+}$, and ${12}^{+}$ states in order from the bottom.

Figure 2

Convergence of the J-VAP VMC energies of ${}^{49}\mathrm{Cr}$ with GXPF1A interaction. The figure shows the energy expectation values of $5/2^{-}$ (filled black circles), $7/2^{-}$ (open blue squares), $9/2^{-}$ (filled orange triangles), and $11/2^{-}$ (open green circles) states, respectively, as functions of the number of iterations. The exact shell-model energies are shown as the rightmost red triangles.

Figure 3

VMC results with the variation after approximate angular-momentum projection against the total number of the mesh points for the integral, $N_z{N}_y$. The filled black symbols connected with the dotted lines denote the converged results of the $I^{\pi }=0^{+}$, $2^{+}$, $4^{+}$, $6^{+}$, $8^{+}$, ${10}^{+}$, and ${12}^{+}$ states of ${}^{48}\mathrm{Cr}$ with the GXPF1A interaction [24]. The open orange symbols connected with the dashed lines denote the fully projected energy of the resultant wave function. See text for further details.

Figure 4

Energy variance extrapolation by the variation after the approximate angular-momentum projection. The $0^{+}$, $2^{+}$, $4^{+}$, and $6^{+}$ shell-model energies of ${}^{48}\mathrm{Cr}$ are obtained with the GXPF1A interaction [24]. The energy expectation values against the energy variance are plotted as the black symbols with the approximate projection. The numbers of the points for the projection are taken as $(N_z,N_y)=(2,1)$, (4,2), (6,3), (8,4), (10,5), and (21,11). The red lines are chi-square-fitted to the symbols. The red squares on the $y$ axis are the exact shell-model energies.