Correlations in eigenfunctions of quantum chaotic systems with sparse Hamiltonian matrices

Area of Science:

• Quantum mechanics
• Statistical physics
• Chaos theory

Background:

• Realistic models of quantum chaotic systems often feature sparse Hamiltonian matrices.
• Understanding eigenfunctions and their correlations is crucial for characterizing quantum chaos.

Purpose of the Study:

• To investigate correlations in eigenfunctions of quantum chaotic systems with sparse Hamiltonians.
• To derive explicit analytical expressions for energy-dependent correlation functions.
• To explore applications of these findings in transition probabilities and local observables.

Main Methods:

• Analysis of Hamiltonian matrices with sparse structures.
• Derivation of analytical expressions for correlation functions.
• Numerical simulations to validate analytical results across various models.

Main Results:

• Explicit expressions for certain energy-dependent correlation functions were derived.
• Analytical results were confirmed through numerical simulations in multiple models.
• The study established a connection between correlation functions and physical observables.

Conclusions:

• The derived analytical expressions provide valuable tools for studying quantum chaotic systems.
• The findings offer a deeper understanding of the behavior of eigenfunctions in sparse Hamiltonian systems.
• Applications in transition probabilities and local observables highlight the practical relevance of the research.