Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 52 (2019)

Low-lying collective excited states in non-integrable pairing models
based on stationary phase approximation to the path integral †F. Ni,∗1 N. Hinohara,∗1,∗2 and T. Nakatsukasa∗1,∗2,∗3

†

∗1
∗2
∗3

Condensed from the article in Phy. Rev. C 98, 064327 (2018)
Faculty of Pure and Applied Sciences, University of Tsukuba
Center for Computational Sciences, University of Tsukuba
RIKEN Nishina Center- 77 -80
B(Pad)The time-dependent mean-field (TDMF) theory isa standard theory to describe the dynamics of nuclei
from the microscopic degrees of freedom.1) Inclusionof the pairing dynamics leads to the time-dependent
Hartree-Fock-Bogoliubov (TDHFB) theory, which has
been utilized for a number of studies of nuclear reaction and structure. The small-amplitude approximation of TDHFB with modern energy density functionals, namely quasiparticle random phase approximation(QRPA), has successfully reproduced the properties ofgiant resonances in nuclei. In contrast, the QRPA description of low-lying quadrupole vibrations is not as
good as that of the giant resonances. A large-amplitude
nature of the quantum shape fluctuation is supposed to
be important for these low-lying collective states.
The TDMF (TDHFB) theory corresponds to an SPA
solution in the path integral formulation. It lacks apart of quantum fluctuation, which is important in large
amplitude dynamics. To introduce the quantum fluctuation based on the TDHFB theory, requantization isnecessary. A simple and straightforward way of requantization is canonical quantization. This is extensively
utilized for collective models in nuclear physics.
In our previous work,2) we studied various requantization methods for the two-level pair model, in order
to investigate the low-lying excited 0+ states. Since thecollectivity is rather low in the pairing motion in nuclei,
the canonical quantization often fails to produce an approximate answer to the exact solution. In contrast,
the stationary phase approximation (SPA) to the path
integral can give quantitative results not only for theexcitation energies but also for the wave functions andtwo-particle-transfer strengths. The quantized statesobtained in the SPA has two advantages: First, thewave functions are given directly in terms of the microscopic degrees of freedom. Second, the restorationof broken symmetries are automatic. However, the applications of SPA have been limited to integrable systems. This is because we need to find separable periodic trajectories on a classical torus. Since nuclearsystems, of course, correspond to non-integrable systems, a straightforward application of the SPA is notpossible.
In this paper, we propose a new approach of SPA
applicable to non-integrable systems, which is basedon the extraction of the one-dimensional (1D) collective coordinate using the adiabatic self-consistent col-60
40200gs-->gs
02+-->02+
gs-->gs++
02 -->02] Exact] ASCC+SPA

188
190
192
Initial mass number of Pb

Fig. 1. Calculated two-neutron transfer matrix elements forPb isotopes.

lective coordinate (ASCC) method.1) Since the 1D system is integrable, the collective subspace can be quantized with the SPA. The optimal degree of freedom associated with a slow collective motion is determined
self-consistently inside the TDHFB space, without any
assumption. Thus, our approach of ASCC+SPA basically consists of two steps: (1) Find a decoupled 1D
collective coordinate of the collective motion of interest.
(2) Apply the SPA to the collective mode.
On the decoupled collective subspace, which is derived by the ASCC, we apply the Einstein-BrillouinKeller (EBK) quantization rule,
p1 dq 1 = 2πk,(1)Ckwhere k is an integer with a unit of h̄ = 1. Then, on
this trajectory Ck , the k-th stationary state is given as asuperposition of the (generalized) Slater determinants:ρ(q, p)dt |q, p⟩ eiT [q,p] ,(2)|ψk ⟩ =Ck

where T [q, p] is the classical action integral. Thismethod is applied to the neutron pairing model for
Pb isotopes. We calculate the two-neutron-transfer
strengths from the ground to the ground and from theexcited to the excited states (Fig. 1). A reasonableagreement is found between the ASCC+SPA and theexact calculations.
References

1) T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, K. Yabana, Rev. Mod. Phys. 88, 045004 (2016).
2) F. Ni, T. Nakatsukasa, Phys. Rev. C 97, 044310 (2018).