Non-ergodic extended regime in random matrix ensembles: insights from eigenvalue spectra

Area of Science:

• Statistical Physics
• Quantum Chaos
• Random Matrix Theory

Background:

• The non-ergodic extended (NEE) regime is a key area of study in physical and random matrix (RM) models.
• NEE is characterized by fractal wavefunctions and long-range spectral correlations.
• Previous research suggested singular-value decomposition (SVD) as a method to reveal NEE through eigenvalue spectra.

Purpose of the Study:

• To apply singular-value decomposition (SVD) to various random matrix (RM) models.
• To demonstrate SVD's capability in both qualitatively identifying the NEE regime and quantitatively determining the ergodic-NEE transition point.
• To explore the potential of SVD in characterizing the NEE regime within a sparse RM model.

Main Methods:

• Application of singular-value decomposition (SVD) to eigenvalue spectra of random matrix models.
• Analysis of SVD results for super-Poissonian behavior indicative of minibands.
• Quantitative assessment of the ergodic-NEE transition point using SVD.

Main Results:

• SVD successfully reveals the non-ergodic extended (NEE) regime across multiple RM models.
• SVD provides a quantitative measure to pinpoint the transition from ergodic to NEE behavior.
• The study suggests the presence of the NEE regime in sparse RM models, identified via SVD.

Conclusions:

• Singular-value decomposition is an effective tool for characterizing the NEE regime in random matrix theory.
• SVD offers a quantitative method to locate critical transitions within these models.
• This work extends the application of SVD to identify NEE in novel sparse random matrix models.