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Boris L Altshuler1, Vladimir E Kravtsov2, Antonello Scardicchio2,3

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This study explores critical properties of the Anderson model using renormalization group methods. It reveals how fractal dimensions evolve with dimensionality, bridging different theoretical frameworks for Anderson transitions.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Quantum Systems

Background:

  • The Anderson model describes electron localization in disordered systems.
  • Understanding critical properties and transitions is crucial for condensed matter physics.
  • Previous work introduced a novel renormalization group (RG) framework for Anderson transitions.

Purpose of the Study:

  • To investigate the dimensional dependence of critical properties in the Anderson model.
  • To analyze the behavior of the beta-function for fractal dimension.
  • To reconcile different theoretical expansions and understand the role of irrelevant exponents.

Main Methods:

  • Utilizing a recently introduced renormalization group (RG) framework.
  • Analyzing the beta-function for fractal dimension in various dimensional limits.
  • Employing expansions around the random regular graph (RRG) results.
  • Investigating the emergence of irrelevant exponents from nonlinear sigma models.

Main Results:

  • Demonstrated a smooth evolution of the fractal dimension's beta-function from d-dimensions to RRG limits.
  • Showed how d-dimensional and RRG expansions can be reconciled.
  • Illustrated the dimensionality dependence of the renormalization group trajectory governed by the irrelevant exponent.
  • Proposed a conjecture for a lower bound on the fractal dimension.

Conclusions:

  • The developed RG framework provides a unified approach to study Anderson transitions across different dimensions.
  • The findings offer insights into the behavior of disordered quantum systems.
  • This work lays the groundwork for future research on many-body and nonequilibrium quantum systems.