Saturation of number variance in embedded random-matrix ensembles

Area of Science:

• Quantum mechanics
• Statistical physics
• Mathematical physics

Background:

• Random matrix theory (RMT) is crucial for understanding complex quantum systems.
• Classical RMT typically assumes stationary correlation functions.
• Noninteracting particle systems offer a simplified model to explore deviations from classical RMT.

Purpose of the Study:

• To investigate fluctuation properties of embedded random matrix ensembles for noninteracting particles.
• To analyze the stationarity of correlation functions in these systems.
• To examine number variance and spacing distributions in relation to correlation lengths.

Main Methods:

• Analysis of embedded random matrix ensembles for two noninteracting particle systems.
• Study of correlation functions, number variance, and spacing distributions.
• Comparison of findings with established results from integrable systems and classical RMT.

Main Results:

• Correlation functions for these ensembles are nonstationary, diverging from classical RMT.
• Spacing distributions follow Poisson statistics, indicating uncorrelated spectra.
• Number variance exhibits linear behavior for short correlation lengths and saturates for large lengths.

Conclusions:

• The observed saturation effects in number variance are demonstrated for the first time in RMT.
• These findings bridge RMT with characteristics previously observed in integrable systems.
• A conjecture is proposed for saturation effects in interacting particle systems within RMT.