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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Disordered Systems

Background:

  • Anderson localization describes the suppression of wave function propagation in disordered systems.
  • Random Regular Graphs (RRGs) offer a simplified model for studying localization in high dimensions.
  • Previous renormalization group (RG) studies have limitations in capturing complex scaling behaviors.

Purpose of the Study:

  • To extend the renormalization group (RG) analysis to Anderson localization on random regular graphs (RRGs).
  • To investigate the validity of the one-parameter scaling hypothesis in this context.
  • To explain the nonmonotonic behavior of physical quantities near the localization transition.

Main Methods:

  • A generalized renormalization group (RG) analysis was applied to the Anderson model on RRGs.
  • The RG equations were analyzed to understand the flow of parameters with changing connectivity.
  • The beta function for the running fractal dimension was derived and analyzed.

Main Results:

  • The one-parameter scaling hypothesis is recovered for large system sizes for both eigenstates and spectral observables.
  • Two terms with differing signs and dependencies in the beta function explain nonmonotonic behaviors.
  • The RG analysis provides a coherent explanation for observed numerical data in Anderson localization on RRGs.

Conclusions:

  • The developed RG theory successfully explains the complex scaling observed in Anderson localization on RRGs.
  • The study reconciles theoretical predictions with numerical findings for disordered systems.
  • This framework offers insights into many-body localization phenomena.