Nicolò Defenu — SISSA Trieste #
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 \(D_{\rm eff}=2d/\sigma\), 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 \(D'_{\rm eff}\) 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 \(\sigma_∗=2−\eta_{\rm SR} \)
(where \(\eta_{\rm SR}\) is the anomalous dimension of the short-range model), and to subsequently control the critical 
behavior of the system for \(\sigma <\sigma_∗\).