Test of finite temperature random-phase approximation on a Lipkin model

We investigate the applicability of the finite temperature random phase approximation (RPA) using a solvable Lipkin model. We show that the finite temperature RPA reproduces reasonably well the temperature dependence of total strength, both for the positive energy (i.e., the excitation) and the negative energy (i.e., the de-excitation) parts. This is the case even at very low temperatures, which may be relevant to astrophysical purposes.

Figure 1

The strength function for the operator $\stackrel{\wedge }{F}=({\stackrel{\wedge }{K}}_{+}+{\stackrel{\wedge }{K}}_{-})/2$ obtained with several methods for $\mathit{VN}/\epsilon =0.5$. (a), (b), and (c) correspond to the temperature of $T/\epsilon =0.05$, 0.5, and 1.0, respectively. The solid line is the exact result for the grand-canonical ensemble, while the dashed and the dotted lines denote the solutions of finite temperature RPA and thermal Hartree-Fock, respectively. The exact result for the canonical ensemble is almost the same as the solid line, and is not shown in the figure.

Figure 2

Same as Fig. 1a, but only with the negative energy part.

Figure 3

The total strength obtained with the several methods as a function of temperature (the upper panel). The meaning of each line is the same as in Fig. 1, except for the thin solid line which denotes the exact result of the canonical ensemble. The lower panel shows the ratio of the total strength to that at zero temperature.

Figure 4

Contribution of the negative energy part to the total strength shown in Fig. 3. The thin and the thick solid lines are indistinguishable on this scale.

Figure 5

Same as Fig. 1, but for a stronger coupling strength, $\mathit{VN}/\epsilon =2.0$ at temperature $T/\epsilon =0.675$.