Critical behavior of the energy gap and its relation with the Berry phase close to the excited state quantum phase transition in the Lipkin model

In our previous work [Phys. Rev. A \textbf{85}, 044102 (2012)], we have studied the Berry phase of the ground state and exited states in the Lipkin model. In this paper, using Hellmann-Feynman theorem, we derive the relation between the energy gap and the Berry phase close to the excited state quantum phase transition (ESQPT) in the Lipkin model. We find that the energy gap is linearly dependent upon the Berry phase close to the ESQPT. As a result, the critical behavior of the energy gap is similar to that of the Berry phase. In addition, we also perform a semiclassical qualitative analysis about the critical behavior of the energy gap.