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Single parameter scaling in one-dimensional localization revisited

Deych1, Lisyansky, Altshuler

  • 1Physics Department, Seton Hall University, South Orange, New Jersey 07052, USA.

Physical Review Letters
|October 4, 2000
PubMed
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This study precisely calculates Lyapunov exponent variance in a 1D Anderson model with Cauchy-distributed energies. A new scaling parameter and analytical criterion for single parameter scaling are discovered, differing from phase randomization.

Area of Science:

  • Condensed matter physics
  • Disordered systems
  • Quantum mechanics

Background:

  • The Anderson model describes electron localization in disordered materials.
  • Lyapunov exponents quantify sensitivity to initial conditions in dynamical systems.
  • Cauchy distribution is a heavy-tailed probability distribution.

Purpose of the Study:

  • To exactly calculate the variance of the Lyapunov exponent in the one-dimensional Anderson model.
  • To identify and analyze a new scaling parameter.
  • To derive an analytical criterion for single parameter scaling.

Main Methods:

  • Exact calculation of Lyapunov exponent variance.
  • Analysis of systems with Cauchy-distributed random site energies.
  • Application to the Kronig-Penney model with random delta-function potentials.

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Main Results:

  • A novel significant scaling parameter was identified.
  • An exact analytical criterion for single parameter scaling was derived.
  • This criterion differs from the conventional phase randomization condition.

Conclusions:

  • The findings provide a new perspective on scaling in disordered systems.
  • The derived criterion offers a more accurate method for analyzing single parameter scaling.
  • The results have implications for understanding electron transport in disordered potentials.