Fixed Points Structure & Effective Fractional Dimension for O(N) Models with Long-Range Interactions

TL;DR

This paper uses functional renormalization group methods to relate long-range O(N) models to short-range models via an effective fractional dimension, predicting critical exponents and fixed points for systems with power-law interactions.

Contribution

It introduces a novel approach to analyze long-range O(N) models by connecting their critical behavior to short-range models through effective fractional dimensions, including new analytical predictions and an improved approximation method.

Findings

Critical exponents can be derived from short-range models at an effective fractional dimension.

Identification of a long-range fixed point branching from the short-range fixed point at a critical decay exponent.

Analytical predictions for the critical exponent ν as a function of σ and N in 2D and 3D.

Abstract

We study O(N) models with power-law interactions by using functional renormalization group methods: we show that both in Local Potential Approximation (LPA) and in LPA' their critical exponents can be computed from the ones of the corresponding short-range O(N) models at an effective fractional dimension. In LPA such effective dimension is given by $D_{eff}=2d/\sigma$, where d is the spatial dimension and $d+\sigma$ is the exponent of the power-law decay of the interactions. In LPA' the prediction by Sak [Phys. Rev. B 8, 1 (1973)] for the critical exponent $\eta$ is retrieved and an effective fractional dimension $D_{eff}'$ is obtained. Using these results we determine the existence of multicritical universality classes of long-range O(N) models and we present analytical predictions for the critical exponent $\nu$ as a function of $\sigma$ and N: explicit results in 2 and 3 dimensions are given. Finally, we propose an improved LPA" approximation to describe the full theory space of the models where both short-range and long-range interactions are present and competing: a long-range fixed point is found to branch from the short-range fixed point at the critical value $\sigma_* = 2-\eta_{SR}$ (where $\eta_{SR}$ is the anomalous dimension of the short-range model), and to subsequently control the critical behavior of the system for $\sigma < \sigma_*$.