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

Excited-State Quantum Phase Transitions in the Anharmonic Lipkin-Meshkov-Glick Model I: Static Aspects