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Critical statistics for non-Hermitian matrices.

A M García-García1, S M Nishigaki, J J M Verbaarschot

  • 1Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794-3800, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2002
PubMed
Summary

We present a generalized matrix ensemble that bridges Hermitian and non-Hermitian systems. This model reveals critical spectral statistics in the weak non-Hermiticity limit, offering insights into open disordered systems.

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Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature.

Physical review. E, Statistical, nonlinear, and soft matter physics·2003

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.

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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.