The $1^-$ strength functions

Figure 5 compares the $1^-$ strength functions of our iterative method (solid line) with the strength functions from Ref. [17] (dashed line). The solid line shows the IS strength calculated when the generated Arnoldi basis is orthogonalized against spurious mode during iteration and dotted line corresponds to similar iterative calculation without orthogonalization. The low-lying state at 0.72MeV, which has a large overlap with the spurious IS $1^-$ mode disappears when the orthogonalization method of Eqs. (22)-(24) is used. Also for the $1^-$ strength function, 100-120 Arnoldi iterations were needed to produce reasonably accurate results, see Fig. 6.

Figure 5: Main panel: the $1^-$ strength functions in $^{132}{\rm Sn}$, calculated using $100$ Arnoldi iterations and with spurious IS mode removed (solid line), and results of the standard RPA from Ref. [17] (dashed line). Dotted line shows results of $140$ Arnoldi iterations without orthogonalization against the spurious IS mode. Inset: same as in the main panel, but for the IV strength functions. All results were calculated for the SLy4 functional.

Figure 6: Similar to Fig. 4 but for the $1^-$ strength functions. The IV strength-function differences were multiplied by the factor of 200fm$^4$.

When no orthogonalization is made against the spurious mode, the obtained excitations contain small components of the spurious mode. This affects the physical part of the IS strength distribution, especially around 20-30MeV. The standard RPA strength function of Ref. [17] has not been corrected for the spuriosity but only the strength of the lowest-lying state that has a large overlap with the spurious IS mode has been omitted. At 20-30MeV, this strength agrees well with our uncorrected strength.

The orthogonalization method improves the convergence of the strength function, because now the 8.3MeV $1^-$ excitation is lowest in energy and thus converges first. Without orthogonalization against the spurious mode, we need 140 iterations instead of 100 to get acceptably converged strength function.