Finite-Temperature Fidelity-Metric Approach to the Lipkin-Meshkov-Glick Model

The fidelity metric has recently been proposed as a useful and elegant approach to identify and characterize both quantum and classical phase transitions. We study this metric on the manifold of thermal states for the Lipkin-Meshkov-Glick (LMG) model. For the isotropic LMG model, we find that the metric reduces to a Fisher-Rao metric, reflecting an underlying classical probability distribution. Furthermore, this metric can be expressed in terms of derivatives of the free energy, indicating a relation to Ruppeiner geometry. This allows us to obtain exact expressions for the (suitably rescaled) metric in the thermodynamic limit. The phase transition of the isotropic LMG model is signalled by a degeneracy of this (improper) metric in the paramagnetic phase. Due to the integrability of the isotropic LMG model, ground state level crossings occur, leading to an ill-defined fidelity metric at zero temperature.