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This study introduces a generalized spectral form factor (GSFF) to capture high-order energy level correlations in complex systems, offering deeper insights into quantum chaos dynamics beyond traditional two-level correlations.

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Area of Science:

  • Quantum Chaos
  • Statistical Mechanics
  • Complex Systems

Background:

  • The spectral form factor (SFF) is vital for understanding energy-level statistics and diagnosing quantum chaos.
  • Existing SFF definitions primarily focus on two-level correlations, limiting comprehensive dynamic analysis.

Purpose of the Study:

  • To extend the definition of the spectral form factor to include high-order correlations.
  • To introduce the generalized spectral form factor (GSFF) for a more complete characterization of chaotic system dynamics.

Main Methods:

  • Defined correlation functions using the standard deviation of energy levels.
  • Applied Fourier transforms to derive the generalized spectral form factor (GSFF).
  • Utilized random matrices as a model system to demonstrate GSFF features.

Main Results:

  • The GSFF is complex, with real and imaginary parts revealing universal dynamics.
  • The real part of the two-level GSFF exhibits a dip-ramp-plateau structure.
  • The imaginary part of the two-level GSFF shows system-size independent convergence in the long-time limit.

Conclusions:

  • The GSFF provides a more comprehensive understanding of chaotic system dynamics than conventional SFF.
  • Both real and imaginary components of GSFF exhibit universal behaviors.
  • The framework is extendable to analyze higher-order correlations, such as three-level GSFF.