Application of random phase approximation to vibrational excitations of double-$\mathrm{\Lambda }$ hypernuclei

Using the Hartree-Fock plus random-phase-approximation (HF+RPA), we study the impurity effect of $\mathrm{\Lambda }$ hyperon on the collective vibrational excitations of double-$\mathrm{\Lambda }$ hypernuclei. To this end, we employ a Skyrme-type $\Lambda N$and$\Lambda\Lambda$ interactions for the HF calculations, and the residual interactions for RPA derived with the same interactions. We find that inclusion of two $\mathrm{\Lambda }$ hyperons in ${}^{16}$O shifts the energy of the collective states towards higher energies. In particular, the energy of the giant monopole resonance of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}^{18}$O, as well as that of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}^{210}$Pb, becomes larger. This implies that the effective incompressibility modulus increases due to the impurity effect of $\mathrm{\Lambda }$ particle, if the $\beta$-stability condition is not imposed.