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

We study, by renormalization group methods, O(N) models with interactions decaying as power law with exponent d+σ. When only the long-range momentum term p(σ) is considered in the propagator, the critical exponents can be computed from those of the corresponding short-range O(N) models at an effective fractional dimension D(eff). Neglecting wave function renormalization effects the result for the effective dimension is D(eff)=2d/σ, which turns to be exact in the spherical model limit (N→∞). Introducing a running wave function renormalization term the effective dimension becomes instead D(eff)=(2-η(SR))d/σ. The latter result coincides with the one found using standard scaling arguments. Explicit results in two and three dimensions are given for the exponent ν. We propose an improved method to describe the full theory space of the models where both short- and long-range propagator terms are present and no a priori choice among the two in the renormalization group flow is done. The eigenvalue spectrum of the full theory for all possible fixed points is drawn and a full description of the fixed-point structure is given, including multicritical long-range universality classes. The effective dimension is shown to be only approximate, and the resulting error is estimated.