Alternative solvable description of the E(5) critical point symmetry in the interacting boson model

A solvable extended Hamiltonian that includes multipair interactions among $s$ and $d$ bosons up to infinite order within the framework of the interacting boson model (IBM) is proposed to gain a better description of E(5) model results for finite-$N$ systems. Numerical fits to low-lying energy levels and reduced $E2$ transition rates within this extended version of the theory are presented for various $N$ values. The fits show that the extended Hamiltonian within the IBM provides a better description of the E(5) model results for small-$N$ cases, while the results of the model in the large-$N$ cases are close to those of the $\mathrm{E}\left(5\right)\text{−}{\beta }^{2n}$ type models studied previously.

Figure 1

The ratio of the parameters $\Delta /\lambda$ used in the fits as a function of $N$.

Figure 2

(a) The potential energy surface (in units of $10g_2$) of the EXT (solid curve) and that derived from the CQ in units of $g=g_2$ (dashed curve) as functions of $\beta$ with $N=10$, where the potential energy is set with $u\left(0\right)=0$. (b) The potential energy surfaces (in units of $g_2$) as functions of $\beta$ for various $N$ cases, in which the curves from bottom to top are those with $N=5$–10, respectively.

Figure 3

(a) The potential energy surface (in units of $g_2$) of the EXT (solid curve) and that derived from the CQ in units of $g=g_2/40$ (dashed curve) as functions of $\beta$ with $N=16$. (b) The second derivative of $u\left(\beta \right)$ at and near $\beta =0$ with the same conditions as indicated for the left panel.