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

Area of Science:

• Quantum Physics
• Condensed Matter Theory
• Geometric Methods in Physics

Background:

• Quantum phase transitions (QPTs) are fundamental phenomena in quantum many-body systems.
• Geometric properties of quantum systems, such as metric tensors and scalar curvatures, offer insights into their behavior.
• The Dicke and Lipkin-Meshkov-Glick models are canonical models exhibiting QPTs.

Purpose of the Study:

• To investigate the classical analog of the quantum metric tensor and scalar curvature.
• To analyze the geometric properties of the parameter space in the Dicke and Lipkin-Meshkov-Glick models.
• To understand how these geometric quantities characterize quantum phase transitions.

Main Methods:

• Calculation of classical and quantum metric tensors.
• Analysis of scalar curvature in the thermodynamic limit.
• Numerical investigation for finite-size systems.

Main Results:

• For the Dicke model, classical and quantum metrics exhibit similar divergence near QPTs, while scalar curvatures diverge only under resonance conditions.
• For the Lipkin-Meshkov-Glick model, classical and quantum metrics agree perfectly; scalar curvature is phase-dependent and approaches a negative constant.
• Finite-size analysis reveals precursors of QPTs in metric and scalar curvature, characterizing them by system parameters and size.

Conclusions:

• The classical analog of quantum geometry provides a powerful tool to study quantum phase transitions.
• Geometric quantities like metric tensors and scalar curvatures can serve as order parameters or indicators of QPTs.
• The behavior of these geometric measures differs between models and depends on system parameters and size.