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

Preprint number: CP3-Origins-2014-22 DNRF90 and DIAS-2014-22
Authors: Alessandro Codello (CP3-Origins & DIAS), Nicolo Defenu (SISSA), and Andrea Trombettoni (SISSA & CNR-IOM DEMOCRITOS & INFN)
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 $$\sigma < \sigma_*$$.