The basic Lipkin-Meshkov-Glick model displays a second order ground state quantum phase transition and an excited state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The new ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility.

It is worth noting that phase transitions are well defined for macroscopic systems, however, the same ideas can be applied when studying mesoscopic systems where one can observe phase transition precursors even for moderate system sizes [30]. Hence, when dealing with mesoscopic systems, the study of their mean field or large-size limit to connect with thermodynamic phase transitions is a valuable reference. Exactly-solvable models, such as the LMG model, are simple enough to be solved for a large number of particles, allowing for a clear connection with the aforementioned large-size limit. This work is part of a more complete study on the anharmonic LMG (aLMG) model. The additional anharmonic term induces, in addition to the already known ESQPT [27,31], an anharmonicity-induced ESQPT that needs to be well understood. In a previous publication [32], the static aspects of both the ground state QPT and the two ESQPT's in the aLMG model were characterized. A mean field analysis in the large-N limit was performed and different observables were used to characterize the different quantum phase transitions involved: the energy gap between adjacent levels, the ground state QPT order parameter, the participation ratio, the quantum fidelity susceptibility, and the level density. In this work, we concentrate on the influence of the two ESQPTs on the dynamics of the aLMG model. With this aim, a quantum quench protocol that consists of an abrupt change in one of the control parameters in the aLMG Hamiltonian is defined. Then, the evolution of the system after the quench is studied using