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Area of Science:

  • Quantum mechanics
  • Random matrix theory
  • Number theory

Background:

  • Understanding quantum system dynamics and symmetries is crucial.
  • Existing methods may not fully capture local spectral properties.

Purpose of the Study:

  • Introduce a novel concept of local level spacings.
  • Develop a random-matrix-theory framework to analyze these spacings.
  • Establish a method to identify quantum system symmetries and dynamics.

Main Methods:

  • Utilize random matrix theory, specifically in the infinite-dimensional limit.
  • Analyze statistics of local level spacings and their ratios.
  • Perform numerical experiments on diverse quantum systems.

Main Results:

  • Determine universal sequences for mean local spacings and their ratios.
  • Demonstrate that these sequences uniquely identify global symmetries.
  • Showcase the ability to distinguish between chaotic and regular quantum dynamics.
  • Corroborate findings with applications to Riemann zeta function zeros, quantum billiards, and Sachdev-Ye-Kitaev models.

Conclusions:

  • Local level spacings provide a powerful new tool for quantum system analysis.
  • This framework offers a unified approach to monitor spectral properties.
  • The method is applicable to both single- and many-body quantum systems.