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Multifractality in a broad class of disordered systems.

Olaf Stenull1

  • 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 25, 2004
PubMed
Summary

This study explores multifractality in disordered systems, revealing infinite critical exponents at the percolation threshold. These exponents, crucial for understanding system behavior, are calculated using advanced field theory methods.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Disordered systems exhibit complex scaling behaviors, particularly at critical points like the percolation threshold.
  • Understanding multifractality requires analyzing the scaling of various statistical moments (cumulants) of dynamical variables.

Purpose of the Study:

  • To investigate multifractality in a wide range of disordered systems, including the diluted x-y model.
  • To determine the critical exponents governing the scaling behavior of cumulant-averaged dynamical variables at the percolation threshold.
  • To establish a connection between these systems and random resistor networks.

Main Methods:

  • Renormalized field theory is employed to analyze scaling behavior.
  • The study focuses on cumulant-averaged dynamical variables, such as spin directions in the x-y model.

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  • Connections to random resistor network models are utilized to calculate exponents.

    • Main Results:

    • The research reveals that each cumulant possesses its own independent critical exponent, leading to an infinite set of exponents.
    • Multifractal exponents are determined up to the two-loop order.
    • It is observed that for specific Hamiltonians, higher cumulant amplitudes can vanish, simplifying the effective multifractal description.

    Conclusions:

    • The findings confirm the presence of infinite critical exponents in the studied disordered systems.
    • The developed theoretical framework allows for the calculation of these exponents, providing deeper insights into critical phenomena.
    • The study highlights the model-dependent nature of multifractality, where simplification is possible under certain conditions.