Low-energy quadrupole collectivity of Sn nuclei in self-consistent calculations with a semi-realistic interaction

Quadrupole collectivity of the lowest-lying states, focusing on $E_x(2^+_1)$ and $B(E2;0^+_1\to 2^+_1)$, have been investigated for the $N=50-82$ Sn nuclei by applying the self-consistent approaches with the semi-realistic interaction M3Y-P6. Both $E_x(2^+_1)$ and $B(E2;0^+_1\to 2^+_1)$ are well reproduced by the spherical Hartree-Fock-Bogolyubov (HFB) plus quasiparticle random-phase approximation (QRPA) calculations in $N\geq 64$, without adjustable parameters. The measured $B(E2)$ values in the neutron-deficient Sn nuclei cast a puzzle. In $54\leq N\leq 62$, the spherical HFB\,+\,QRPA calculations give too strong $B(E2)$, opposite to the shell-model predictions within the one major shell. Via the constrained-HFB (CHFB) calculations, it is found that the neutron-deficient Sn nuclei are soft against the quadrupole deformation, accounting for the limited applicability of the HFB\,+\,QRPA approach. In particular, the potential energy curves (PECs) are almost flat in the range of $|q_0|\lesssim 200\,\mathrm{fm}^2$ in $^{106-110}$Sn. We confirm that the near degeneracy of $n0g_{7/2}$ and $n1d_{5/2}$ triggers weak quadrupole deformation and its balance with the pairing makes PECs flat, which is qualitatively consistent with a recent shell model result in an extended model space, by the calculations shifting the single-particle energy spacing and the pairing strength. These conclusions are supported by the proton-to-neutron ratios of the transition matrix elements and the reference values of $B(E2)$ with the angular-momentum projection on top of the CHFB solutions.