Nonergodic extended states in the β ensemble.

Area of Science:

• Physics
• Quantum Mechanics
• Statistical Mechanics

Background:

• Matrix models are crucial for understanding many-body localization (MBL).
• The β ensemble is a simple matrix model, but its eigenvector properties are understudied.
• Energy level correlations in the β ensemble are well-researched, unlike eigenvector behaviors.

Purpose of the Study:

• To numerically investigate the eigenvector properties of the β ensemble.
• To identify the conditions for Anderson transitions and ergodicity breakdown in the β ensemble.
• To compare the β ensemble's behavior with other models like the Rosenzweig-Porter ensemble (RPE).

Main Methods:

• Numerical simulations of the β ensemble.
• Analysis of spectral statistics and eigenvector properties.
• Parameter variation to study transitions (β=N^{-γ}).

Main Results:

• The Anderson transition occurs at γ=1 and ergodicity breaks down at γ=0.
• Nonergodic extended (NEE) states are observed for 0<γ<1.
• The chaotic-integrable transition aligns with ergodicity breaking in the β ensemble, unlike in RPE or 1D disordered spin-1/2 Heisenberg models.

Conclusions:

• The β ensemble provides another example of NEE states, distinct from RPE.
• Dynamical timescales in the NEE regime of the β ensemble exhibit unique behavior.
• Understanding these transitions is key for MBL research.