Critical statistics for non-Hermitian matrices

Area of Science:

• Quantum mechanics and statistical physics
• Random matrix theory

Background:

• Non-Hermitian systems exhibit unique spectral properties distinct from Hermitian counterparts.
• Understanding spectral correlations is crucial for characterizing complex quantum systems.

Purpose of the Study:

• Introduce a generalized ensemble of non-Hermitian matrices.
• Investigate spectral correlations in both weak and strong non-Hermiticity regimes.
• Explore potential applications in open disordered systems near Anderson transitions.

Main Methods:

• Extension of the Itzykson-Zuber formula for general complex matrices to derive joint eigenvalue distributions.
• Analysis of correlation functions in the limits of weak and strong non-Hermiticity.
• Comparison with established ensembles like Gaussian Unitary Ensemble, Ginibre ensemble, and Poisson ensemble.

Main Results:

• In the weak non-Hermiticity limit, bulk spectral correlations exhibit critical statistics, approaching linear behavior of number variance at eigenvalue spacing scale.
• The slope of this linear behavior is largely independent of the degree of non-Hermiticity.
• In the strong non-Hermiticity limit, a crossover from Ginibre to Poisson ensemble behavior is observed near the spectrum's surface.

Conclusions:

• The generalized ensemble provides a unified framework for studying spectral properties across different universality classes.
• The observed critical statistics in the weak non-Hermiticity regime offer new insights into spectral behavior.
• The model's relevance to open disordered systems suggests potential for describing Anderson transitions.