Second random-phase approximation, Thouless' theorem, and the stability condition reexamined and clarified

It has been revealed through numerical calculations that the second random phase approximation (SRPA) with the Hartree-Fock solution as its reference state results in 1. spurious states at genuinely finite energy, contrary to common expectation, and 2. unstable solutions, which within the first-order random phase approximation correspond to real low-energy collective vibrations. In the present work, these shortcomings of SRPA are shown to not contradict Thouless' theorem about the energy-weighted sum rule, and their origin is traced to the violation of the stability condition. A more general theorem is proven. Formal arguments are elucidated through numerical examples. Implications for the validity of the SRPA are discussed.

Figure 1

The lowest eigenvalue of the $A\pm B$ matrices as a function of the 2p2h energy cutoff, up to exhaustion of the 2p2h space, for the $1^{-}$ channel of ${}^{16}\mathrm{O}$ within the SRPA (see text).

Figure 2

The energy-weighted sums and spurious-state-only contributions for various dipole operators defined in the text, as a function of the 2p2h energy cutoff, up to exhaustion of the 2p2h space, for the $1^{-}$ channel of ${}^{16}\mathrm{O}$ within the SRPA(0) (see text). (a) Isovector operators. (b) Isoscalar operators.

Figure 3

The non-energy-weighted sums and spurious-state-only contributions for various dipole operators defined in the text, as a function of the 2p2h energy cutoff, up to exhaustion of the 2p2h space, for the $1^{-}$ channel of ${}^{16}\mathrm{O}$ within the SRPA(0) (see text). (a) Isovector operators. (b) Isoscalar operators. Lines between $E_{2p2h,max}=60$ and 80 MeV are drawn only to guide the eye. The discontinuity is caused by the presence of an imaginary solution for low cutoffs (and in the RPA).

Figure 4

(a) Number of 2p2h configurations as a function of the cutoff energy $E_{2p2h,max}$, up to exhaustion of the 2p2h space, and (b) lowest eigenvalue of the $A\pm B$ and $A$ matrices for the $2^{+}$ channel of ${}^{48}\mathrm{Ca}$ within the SRPA (see text). The dotted vertical line marks the first 2p2h configuration. For the meaning of the gray vertical lines, see the text.

Figure 5

(a) Deviations from the RPA energy-weighted sums and (b) behavior of the SRPA eigenvalues, $\pm e_1$ and $\pm e_2$, lying closest to zero, as a function of the 2p2h energy cutoff for the $2^{+}$ channel of ${}^{48}\mathrm{Ca}$ within the SRPA (see text). In (b) the absolute values of $\Re e_{1,2}$ and $\Im e_{1,2}$ are plotted, but additional information (e.g., sign of positive-norm eigenenergies, if they are real) is given explicitly for each $E_{2p2h,max}$ domain, marked by vertical gray lines. The dotted vertical line marks the energy of the first 2p2h configuration.