Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Area of Science:

• Quantum mechanics
• Statistical physics
• Random matrix theory

Background:

• Complex quantum systems often exhibit critical statistics in their spectral properties.
• Random matrix theory provides a framework for studying these spectral correlations.
• The Calogero-Sutherland (CS) model describes interacting particles and has connections to spectral properties.

Purpose of the Study:

• To introduce and analyze a generalized Gaussian orthogonal ensemble for critical statistics.
• To establish a connection between this random matrix model and the CS Hamiltonian.
• To investigate the spectral correlation properties, specifically number variance and Delta(3) statistics.

Main Methods:

• Expressing the joint eigenvalue distribution using the density matrix of the CS model.
• Utilizing a conjectured expression for density-density correlations in the CS model.
• Analytical calculation of number variance and Delta(3) statistics for the random matrix model.

Main Results:

• The number variance of the random matrix model is asymptotically linear.
• The slope of the linear number variance depends on model parameters, indicating critical statistics.
• Analytical results accurately describe the number variance and Delta(3) statistic for specific quantum systems.

Conclusions:

• The developed random matrix model accurately captures critical spectral statistics.
• This model is applicable to complex quantum systems with fractal classical counterparts.
• The findings provide insights into spectral correlations in systems like the anisotropic Kepler problem and kicked particle potentials.