Renormalization group for Anderson localization on high-dimensional lattices.

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.