Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Statistical Mechanics

Background:

• Multifractals describe scaling in diverse natural systems using a spectrum of exponents.
• Anderson transitions involve critical wave functions exhibiting multifractality.
• Field theories describe these transitions using operators with specific scaling dimensions.

Purpose of the Study:

• To investigate the mathematical properties of multifractal spectra in Anderson transitions.
• To derive constraints on the multifractal spectrum using theoretical frameworks.
• To establish a fundamental relationship for multifractality in critical phenomena.

Main Methods:

• Utilizing the conformal bootstrap framework.
• Assuming conformal invariance and Abelian fusion properties of operators.
• Applying operator product expansion techniques.

Main Results:

• A constraint on the multifractal spectrum (Δ_{q}) was derived.
• The multifractal spectrum and its generalized form are shown to be quadratic.
• This result holds for any dimension d≥2.

Conclusions:

• The quadratic nature of the multifractal spectrum is a fundamental consequence of conformal invariance and Abelian fusion.
• This finding provides a simplified and universal description of multifractality in Anderson transitions.
• The conformal bootstrap offers a powerful tool for constraining scaling dimensions in critical systems.