Beyond mean field study of excited states: Analysis within the Lipkin model

We compare the generator coordinate method (GCM) and the random-phase approximation (RPA) in the framework of the exactly solvable Lipkin-Meshkov-Glick model. We show that the discretized GCM works quite well and permits to obtain results close to the exact results with a small number of discretization points.

Figure 1

Mean-field potentials for two values of the interaction strength, $\chi =0.4$ and $\chi =1.2$, for $N=30$ particles.

Figure 2

Comparison between the exact and GCM energies of the first three states of the LMG model for 30 particles. The solid lines are the exact result, while the dashed and dotted lines represent the GCM solutions obtained with the two different meshes (see text) with seven points in α. The HF results are indicated by a dash-dotted line.

Figure 3

Comparison between the exact and GCM energies of the first three states of the LMG model for a system with $N=50$. Two generator coordinates, α and φ are used in the GCM. The exact result are represented by solid lines. The dotted and dashed lines lines are GCM results with seven and nine points in α; the dash-dotted lines are results of calculations with seven points in α and three points in φ.

Figure 4

Excitation energies of the first two excited states as a function of the interaction strength χ. Exact results are given by solid lines, the RPA results by dotted lines. The dashed and dash-dotted lines correspond to GCM calculations with the same choices of points as in Fig. 2.