Long-range level correlations in quantum systems with finite Hilbert space dimension

Area of Science:

• Quantum mechanics
• Statistical physics
• Condensed matter theory

Background:

• Spectral statistics are crucial for understanding quantum systems.
• Uncorrelated energy levels are a key assumption in some models.
• Finite Hilbert spaces present unique challenges for spectral analysis.

Purpose of the Study:

• To investigate the global correlation of eigenlevels in quantum systems with finite Hilbert spaces.
• To analyze the impact of spectral unfolding procedures on statistical measures.
• To develop a model explaining observed phenomena in spectral statistics.

Main Methods:

• Derivation of a theorem on eigenlevel correlations.
• Analytic and numerical calculation of power spectrum for the δn statistic.
• Testing a proposed model on the disordered XXZ chain and the Gaudin-Richardson model.

Main Results:

• Eigenlevels in finite quantum systems are globally correlated, irrespective of integrability.
• The unfolding procedure can distort spectral statistics, invalidating analytic results.
• A simple model accurately accounts for the observed effects of unfolding.

Conclusions:

• The assumption of globally uncorrelated eigenlevels is not universally valid for finite quantum systems.
• Careful consideration of spectral unfolding is essential for accurate analysis of quantum system dynamics.
• The developed model provides a framework for understanding spectral statistics in complex quantum systems.