Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models

TL;DR

This paper investigates the classical analogs of quantum metric tensors and scalar curvatures in the Dicke and Lipkin-Meshkov-Glick models, revealing their behavior near quantum phase transitions and their finite-size precursors.

Contribution

It provides a detailed analysis of the classical and quantum geometric properties of these models, highlighting their divergence behavior and phase-dependent scalar curvature characteristics.

Findings

Classical and quantum metrics agree in the Lipkin-Meshkov-Glick model.

Divergent behavior of metrics near quantum phase transitions.

Finite-size analysis reveals precursors of phase transitions.

Abstract

We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size.