Fotoc complexity in the Lipkin–Meshkov–Glick model and its variant

We study fidelity out-of-time-order correlators (FOTOCs) in the Lipkin–Meshkov–Glick (LMG) model and one of its variants dubbed as the extended LMG model. We demonstrate that these exhibit distinctive behaviour at quantum phase transitions (QPTs) in both the ground and the excited states. We show that the dynamics of the FOTOC have different behaviour in the symmetric and broken phases and as one approaches QPT. Rescaling the FOTOC operator with time, we establish that for small times the rescaled operator is identical to the Loschmidt echo. We also compute the Nielsen complexity of the FOTOC operator for both models and apply this operator on the ground and excited states to obtain the quasi-scrambled states. The FOTOC operator introduces a small perturbation on the original ground and excited states. For this perturbed state, we compute the quantum information metric to first and second order in perturbation, in the thermodynamic limit. We find that the associated Ricci scalar does not show any divergence or discontinuity at QPTs in the LMG model. Instead, the amplitude of oscillations is relatively lower in the symmetric phase than in the broken phase. However, in the case of extended LMG model, the Ricci scalar diverges at the QPT from the broken phase side, in contrast to the zeroth order result. Finally, we comment upon the Fubini-Study complexity in the extended LMG model. Graphical abstract