Collapse of the random-phase approximation: Examples and counter-examples from the shell model

The Hartree-Fock approximation to the many-fermion problem can break exact symmetries, and in some cases by changing a parameter in the interaction one can drive the Hartree-Fock minimum from a symmetry-breaking state to a symmetry-conserving state (also referred to as a “phase transition” in the literature). The order of the transition is important when one applies the random-phase approximation (RPA) to the of the Hartree-Fock wave function: if first order, RPA is stable through the transition, but if second-order, then the RPA amplitudes become large and lead to unphysical results. The latter is known as “collapse” of the RPA. While the difference between first- and second-order transitions in the RPA was first pointed out by Thouless, we present for the first time nontrivial examples of both first- and second-order transitions in a uniform model, the interacting shell-model, where we can compare to exact numerical results.

Figure 1

Sketches of “phase” transitions in the energy landscape. (a) A first-order transition. (b) A second-order transition.

Figure 2

Ground-state energy of ${}^{28}\mathrm{Si}$, calculated in the full interacting shell model (SM, solid lines), Hartree-Fock (HF, dotted lines), and Hartree-Fock plus random-phase approximation (HF$+$RPA, dashed lines). Here Δ is added to the $1s_{1/2}$ and $0d_{3/2}$ single-particle energies; $\Delta =0$ corresponds to the original Wildenthal values. The left-hand panel shows only the final result; the right-hand panel identifies spherical and (oblate) deformed mean-field phases. Note the significant region of coexistence.

Figure 3

For ${}^{28}\mathrm{Si}$. (Upper panel) Low-lying RPA frequencies Ω for spherical (dashed) and deformed (solid) HF states. (Lower panel) $\left|Y_{\lambda }\right|{}^2$ corresponding to the RPA frequencies in the upper panel. In both cases the degeneracy is given in parentheses; for the spherical case the degeneracy $=2J+1$ where $J$ is the angular momentum of the RPA mode.

Figure 4

Similar to Fig. 2 but for ${}^{32}\mathrm{S}$; here Δ is the change in the $0d_{3/2}$ energy.

Figure 5

Similar to Fig. 3 but for ${}^{32}\mathrm{S}$; here Δ is the change in the $0d_{3/2}$ energy.

Figure 6

Ground-state energy of ${}^{12}\mathrm{C}$ in the $0p_{1/2}\text{−}0p_{1/2}\text{−}0d_{5/2}\text{−}1s_{1/2}$ space, calculated in the full interacting shell model (SM) (solid line), Hartree-Fock (HF) (dotted line), and Hartree-Fock plus random-phase approximation (RPA) (dashed line). Here Δ is added to the $1s_{1/2}$ and $0d_{5/2}$ single-particle energies; $\Delta =0$ puts the negative-parity states in the SM at approximately the correct location. We show where the HF states and corresponding HF$+$RPA states have either exact parity (and also spherical symmetry) or mixed parity (and break rotational invariance as well).

Figure 7

For ${}^{12}\mathrm{C}$. (Upper panel) low-lying RPA frequencies Ω for exact-parity and parity-mixing HF states. (Lower panel) $\left|Y_{\lambda }\right|{}^2$ corresponding to the RPA frequencies in the upper panel. In both cases the degeneracy is given in parentheses; the exact-parity HF states are oblate, allowing for time-reversed pairs, but for parity-mixing the HF state becomes triaxial, breaking the time-reversal degeneracy.