Quantum geometric tensor and quantum phase transitions in the Lipkin-Meshkov-Glick model

We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground-state quantum phase transition where a bifurcation occurs, showing a change in stability associated with an excited-state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest-energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find excellent agreement between them for large sizes of the system in a wide region of the parameter space except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid.

Automated summary

This study investigates the quantum metric tensor and scalar curvature of a specific version of the Lipkin-Meshkov-Glick model, analyzing ground-state and excited-state quantum phase transitions. By utilizing the Holstein-Primakoff approximation, analytic expressions for quantum metric tensor and curvatures were derived and compared with finite-size numerical results, showing good agreement except near phase transition points. The classical Hamiltonian was constructed using Bloch coherent states, revealing stability changes and bifurcation during quantum phase transitions.