Random Matrix Ensemble for the Level Statistics of Many-Body Localization

Area of Science:

• Quantum chaos
• Statistical mechanics
• Condensed matter physics

Background:

• Wigner-Dyson statistics describe energy level distributions in quantum systems.
• The Gaussian β ensemble generalizes these statistics to a continuous range of β.
• Many-body localization (MBL) describes a transition from thermal to localized behavior in interacting quantum systems.

Purpose of the Study:

• To numerically investigate the level statistics of the Gaussian β ensemble.
• To establish the physical relevance of these statistics in the context of many-body localization.
• To explore the behavior of the Gaussian β ensemble for Hamiltonians with broken time-reversal symmetry.

Main Methods:

• Numerical simulations of the Gaussian β ensemble.
• Comparison of level statistics with a paradigmatic MBL model.
• Analysis of a related Hamiltonian with broken time-reversal symmetry.

Main Results:

• The Gaussian β ensemble provides a smooth interpolation between Poissonian (β→0) and Wigner-Dyson (β=1, 2, 4) level statistics.
• Excellent agreement was found between the Gaussian β ensemble statistics and the MBL model across the thermal-to-localized phase transition.
• Similar agreement was observed for a Hamiltonian with broken time-reversal symmetry.

Conclusions:

• The Gaussian β ensemble is physically relevant for describing level statistics in complex quantum systems.
• These statistics accurately capture the behavior of systems undergoing many-body localization.
• The framework extends to systems with broken time-reversal symmetry, broadening its applicability.