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

TL;DR

This paper investigates the quantum geometric tensor and phase transitions in the Lipkin-Meshkov-Glick model, revealing how ground and excited states undergo bifurcations and stability changes, with analytic and numerical methods showing strong agreement.

Contribution

It introduces an analytic approach using the Holstein-Primakoff approximation to study quantum geometry and phase transitions in the LMG model, validated by numerical results.

Findings

Quantum phase transition characterized by bifurcation in stationary points.

Analytic expressions for quantum metric tensor and curvatures derived.

Excellent agreement between analytic and numerical results away from critical points.

Abstract

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 of 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 an 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.