The Effects of Multi-$\Lambda$ Hyperons on Collective Modes in Nuclei

The dynamical influence of $\Lambda$ hyperons on the excited-state properties of closed-shell multi-$\Lambda$ Ca, Ni, Sn and Pb hypernuclei is investigated using the self-consistent Hartree-Fock + Random Phase Approximation in coordinate space. The strength distributions for the isoscalar monopole, isovector dipole, and isoscalar quadrupole modes are calculated, revealing a systematic upward energy shift with increasing $\Lambda$ hyperon number $-S$. The scaling behavior of the computed centroid energies $\sqrt{m_1/m_{-1}}$ with respect to both the mass and hyperon number is determined. The nuclear incompressibility modulus $K_A$ is found to increase monotonically with $-S$. The largest value is found in the $^{258}_{50\Lambda}$Pb hypernucleus, reaching $K_A = 322$ MeV. Calculations in uniform hypernuclear matter confirm that this stiffening is a bulk effect driven by both the $N\Lambda$ and $\Lambda\Lambda$ interactions. Analysis of the transition densities for states with maximal collective coherence indicates that the dynamical effect of $\Lambda$ hyperons is predominantly in phase with the protons, especially in the case of the isovector E1 modes.