Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions.

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Quantum Field Theory

Background:

• O(N) models are fundamental in statistical mechanics and quantum field theory.
• Interactions decaying as power laws introduce long-range correlations.
• Renormalization group methods are crucial for understanding critical phenomena.

Purpose of the Study:

• To analyze O(N) models with power-law decaying interactions using renormalization group methods.
• To compute critical exponents by introducing an effective fractional dimension.
• To explore the full theory space including both short- and long-range propagator terms.

Main Methods:

• Renormalization group (RG) analysis.
• Calculation of critical exponents.
• Investigation of effective fractional dimensions.
• Analysis of propagator terms and wave function renormalization.

Main Results:

• An effective fractional dimension D(eff) is derived for O(N) models with power-law interactions.
• The effective dimension is D(eff)=2d/σ neglecting wave function renormalization, exact in the spherical model limit.
• Including wave function renormalization yields D(eff)=(2-η(SR))d/σ, matching standard scaling arguments.
• Explicit results for the exponent ν in two and three dimensions are provided.
• A comprehensive analysis of the fixed-point structure and multicritical long-range universality classes is presented.

Conclusions:

• The effective dimension is an approximation, and its error is estimated.
• The proposed method offers a complete description of the theory space for these models.
• This work advances the understanding of critical phenomena in systems with long-range interactions.