Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems.

Area of Science:

• Quantum mechanics
• Statistical physics
• Complex systems

Background:

• Multifractal dimensions are crucial for understanding quantum state localization in complex systems.
• Ergodic states in large systems approach fractal dimensions of unity, but this convergence is slow.
• Understanding scaling and finite-size effects of fractal dimensions is essential for accurate characterization.

Purpose of the Study:

• To investigate the scaling and finite-size properties of multifractal dimensions.
• To compare multifractal dimension behavior in random matrix ensembles with chaotic quantum systems.
• To assess the ergodicity of local many-body Hamiltonians.

Main Methods:

• Analytical derivation of finite-size dependence for multifractal dimensions in random matrix ensembles.
• Numerical simulations of the quantized standard map (kicked rotor) and a XXZ spin chain.
• Comparison of multifractal dimension scaling across different quantum systems.

Main Results:

• Random matrix ensembles provide an analytical lower bound for multifractal dimensions.
• Finite sample sizes significantly impact random matrix computations, requiring exponentially large datasets for accuracy.
• The kicked rotor closely matches random matrix predictions, including finite statistics effects.
• XXZ spin chains show system-specific scaling, deviating from random matrix theory, indicating weak ergodicity.

Conclusions:

• Random matrix theory provides a baseline but has limitations for certain quantum systems.
• The kicked rotor demonstrates characteristics consistent with random matrix predictions.
• Local many-body Hamiltonians, like the XXZ spin chain, may exhibit 'weak ergodicity', deviating from standard random matrix predictions.