A review of sigma models for quantum chaotic dynamics

Area of Science:

• Quantum mechanics
• Statistical physics
• Mathematical physics

Background:

• Supersymmetric sigma models offer a powerful framework for studying complex quantum systems.
• Unitary maps are crucial for describing time evolution in quantum mechanics.
• Quantum chaos investigates the quantum mechanical behavior of systems with chaotic classical counterparts.

Purpose of the Study:

• To review the construction of the supersymmetric sigma model for unitary maps.
• To apply this model to understand spectral fluctuations in quantum chaotic systems.
• To differentiate conditions leading to universal fluctuations versus quantum localization.

Main Methods:

• Utilizing the color-flavor transformation for model construction.
• Analyzing three distinct case studies: general Floquet maps, quantum graphs, and the kicked rotor.
• Investigating the relationship between classical dynamics (ergodicity, decay rates) and quantum spectral properties.

Main Results:

• Universal spectral fluctuations are demonstrated in general Floquet maps and quantum graphs under fully chaotic classical dynamics.
• The kicked rotor model exhibits diffusion and long-lived excitation modes, preventing universal fluctuations.
• Quantum localization is shown to occur in the kicked rotor system.

Conclusions:

• The supersymmetric sigma model provides insights into spectral fluctuations in quantum chaos.
• The nature of classical dynamics (fully chaotic vs. diffusive) dictates the emergence of universal spectral fluctuations or quantum localization.
• This work highlights the importance of system-specific classical dynamics in determining quantum chaotic behavior.