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

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 $D_{eff}=2d/σ$, where d is the spatial dimension and $d+σ$ 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 $η$ 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 $ν$ as a function of $σ$ 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 $σ_* = 2-η_{SR}$ (where $η_{SR}$ is the anomalous dimension of the short-range model), and to subsequently control the critical behavior of the system for $σ< σ_*$.