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Multifractality and intermediate statistics in quantum maps.

J Martin1, O Giraud, B Georgeot

  • 1Laboratoire de Physique Théorique, Université de Toulouse, UPS, CNRS, Toulouse, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 4, 2008
PubMed
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We explored multifractal properties of quantum wave functions, linking their fractal dimensions to spectral statistics. This finding offers insights into quantum chaos and transitions like Anderson and quantum Hall effects.

Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Complex systems

Background:

  • Wave functions in quantum systems can exhibit complex, fractal properties.
  • Spectral statistics categorize quantum systems as either integrable or chaotic.
  • Multifractality is observed in various physical phenomena, including disordered systems.

Purpose of the Study:

  • To investigate the multifractal properties of wave functions in a family of quantum maps.
  • To establish a relationship between multifractality and spectral statistics.
  • To explore potential applications in condensed matter physics.

Main Methods:

  • Numerical computations of wave function multifractality.
  • Analytical derivation of generalized fractal dimensions.

Related Experiment Videos

  • Analysis of a one-parameter family of quantum maps spanning integrable to chaotic regimes.

    • Main Results:

    • Generalized fractal dimensions are directly correlated with the parameter of the classical map.
    • This parameter also governs the spectral statistics of the quantum system.
    • A direct link is established between wave function multifractality and spectral statistics.

    Conclusions:

    • The multifractal properties of wave functions are quantitatively related to spectral statistics.
    • Findings are relevant for understanding Anderson and quantum Hall transitions.
    • The study provides a framework for analyzing multifractality in quantum chaotic systems.